Algebra Examples

$\tan (x - y) = \frac{y}{1 + x^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\tan (x - y)) = \frac{d}{d y} (\frac{y}{1 + x^{2}})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \tan (y)$ and $g (y) = x - y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x - y$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d y} [x - y]$

Step 2.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d y} [x - y]$

Step 2.1.3

Replace all occurrences of $u$ with $x - y$ .

$\sec^{2} (x - y) \frac{d}{d y} [x - y]$

Step 2.2

By the Sum Rule, the derivative of $x - y$ with respect to $y$ is $\frac{d}{d y} [x] + \frac{d}{d y} [- y]$ .

$\sec^{2} (x - y) (\frac{d}{d y} [x] + \frac{d}{d y} [- y])$

Step 2.3

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\sec^{2} (x - y) (x' + \frac{d}{d y} [- y])$

Step 2.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

$\sec^{2} (x - y) (x' - \frac{d}{d y} [y])$

Step 2.5

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\sec^{2} (x - y) (x' - 1 \cdot 1)$

Step 2.6

Multiply $- 1$ by $1$ .

$\sec^{2} (x - y) (x' - 1)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = y$ and $g (y) = 1 + x^{2}$ .

$\frac{(1 + x^{2}) \frac{d}{d y} [y] - y \frac{d}{d y} [1 + x^{2}]}{{(1 + x^{2})}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\frac{(1 + x^{2}) \cdot 1 - y \frac{d}{d y} [1 + x^{2}]}{{(1 + x^{2})}^{2}}$

Step 3.2.2

Multiply $1 + x^{2}$ by $1$ .

$\frac{1 + x^{2} - y \frac{d}{d y} [1 + x^{2}]}{{(1 + x^{2})}^{2}}$

Step 3.2.3

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [x^{2}]$ .

$\frac{1 + x^{2} - y (\frac{d}{d y} [1] + \frac{d}{d y} [x^{2}])}{{(1 + x^{2})}^{2}}$

Step 3.2.4

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{1 + x^{2} - y (0 + \frac{d}{d y} [x^{2}])}{{(1 + x^{2})}^{2}}$

Step 3.2.5

Add $0$ and $\frac{d}{d y} [x^{2}]$ .

$\frac{1 + x^{2} - y \frac{d}{d y} [x^{2}]}{{(1 + x^{2})}^{2}}$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x$ .

$\frac{1 + x^{2} - y (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x])}{{(1 + x^{2})}^{2}}$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{1 + x^{2} - y (2 u \frac{d}{d y} [x])}{{(1 + x^{2})}^{2}}$

Step 3.3.3

Replace all occurrences of $u$ with $x$ .

$\frac{1 + x^{2} - y (2 x \frac{d}{d y} [x])}{{(1 + x^{2})}^{2}}$

Step 3.4

Multiply $2$ by $- 1$ .

$\frac{1 + x^{2} - 2 y (x \frac{d}{d y} [x])}{{(1 + x^{2})}^{2}}$

Step 3.5

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{1 + x^{2} - 2 y x x'}{{(1 + x^{2})}^{2}}$

Step 3.6

Reorder terms.

$\frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$\sec^{2} (x - y) (x' - 1) = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}}$

Step 5

Solve for $x'$ .

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Step 5.1

Multiply both sides by ${(x^{2} + 1)}^{2}$ .

$\sec^{2} (x - y) (x' - 1) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\sec^{2} (x - y) (x' - 1) {(x^{2} + 1)}^{2}$ .

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Step 5.2.1.1.1

Simplify by multiplying through.

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Step 5.2.1.1.1.1

Apply the distributive property.

$(\sec^{2} (x - y) x' + \sec^{2} (x - y) \cdot - 1) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2.1.1.1.2

Move $- 1$ to the left of $\sec^{2} (x - y)$ .

$(\sec^{2} (x - y) x' - 1 \cdot \sec^{2} (x - y)) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2.1.1.2

Rewrite $- 1 \sec^{2} (x - y)$ as $- \sec^{2} (x - y)$ .

$(\sec^{2} (x - y) x' - \sec^{2} (x - y)) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2.1.1.3

Simplify by multiplying through.

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Step 5.2.1.1.3.1

Apply the distributive property.

$\sec^{2} (x - y) x' {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2.1.1.3.2

Reorder factors in $\sec^{2} (x - y) x' {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2}$ .

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$ .

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Step 5.2.2.1.1

Cancel the common factor of ${(x^{2} + 1)}^{2}$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = \frac{x^{2} - 2 x y x' + 1}{{(x^{2} + 1)}^{2}} {(x^{2} + 1)}^{2}$

Step 5.2.2.1.1.2

Rewrite the expression.

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = x^{2} - 2 x y x' + 1$

Step 5.2.2.1.2

Simplify the expression.

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Step 5.2.2.1.2.1

Move $y$ .

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = x^{2} - 2 x x' y + 1$

Step 5.2.2.1.2.2

Move $x$ .

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = x^{2} - 2 x' x y + 1$

Step 5.2.2.1.2.3

Reorder $x^{2}$ and $- 2 x' x y$ .

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} = - 2 x' x y + x^{2} + 1$

Step 5.3

Solve for $x'$ .

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Step 5.3.1

Move all terms containing $x'$ to the left side of the equation.

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Step 5.3.1.1

Add $2 x' x y$ to both sides of the equation.

$x' \sec^{2} (x - y) {(x^{2} + 1)}^{2} - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2

Simplify each term.

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Step 5.3.1.2.1

Rewrite ${(x^{2} + 1)}^{2}$ as $(x^{2} + 1) (x^{2} + 1)$ .

$x' \sec^{2} (x - y) ((x^{2} + 1) (x^{2} + 1)) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.2

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 5.3.1.2.2.1

Apply the distributive property.

$x' \sec^{2} (x - y) (x^{2} (x^{2} + 1) + 1 (x^{2} + 1)) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.2.2

Apply the distributive property.

$x' \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 1 + 1 (x^{2} + 1)) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.2.3

Apply the distributive property.

$x' \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.3

Simplify and combine like terms.

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Step 5.3.1.2.3.1

Simplify each term.

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Step 5.3.1.2.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.1.2.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x' \sec^{2} (x - y) (x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.3.1.1.2

Add $2$ and $2$ .

$x' \sec^{2} (x - y) (x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.3.1.2

Multiply $x^{2}$ by $1$ .

$x' \sec^{2} (x - y) (x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.3.1.3

Multiply $x^{2}$ by $1$ .

$x' \sec^{2} (x - y) (x^{4} + x^{2} + x^{2} + 1 \cdot 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.3.1.4

Multiply $1$ by $1$ .

$x' \sec^{2} (x - y) (x^{4} + x^{2} + x^{2} + 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.3.2

Add $x^{2}$ and $x^{2}$ .

$x' \sec^{2} (x - y) (x^{4} + 2 x^{2} + 1) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.4

Apply the distributive property.

$x' \sec^{2} (x - y) x^{4} + x' \sec^{2} (x - y) (2 x^{2}) + x' \sec^{2} (x - y) \cdot 1 - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.5

Simplify.

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Step 5.3.1.2.5.1

Rewrite using the commutative property of multiplication.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) \cdot 1 - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.5.2

Multiply $x'$ by $1$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) {(x^{2} + 1)}^{2} + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.6

Rewrite ${(x^{2} + 1)}^{2}$ as $(x^{2} + 1) (x^{2} + 1)$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) ((x^{2} + 1) (x^{2} + 1)) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.7

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 5.3.1.2.7.1

Apply the distributive property.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{2} (x^{2} + 1) + 1 (x^{2} + 1)) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.7.2

Apply the distributive property.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 1 + 1 (x^{2} + 1)) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.7.3

Apply the distributive property.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.8

Simplify and combine like terms.

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Step 5.3.1.2.8.1

Simplify each term.

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Step 5.3.1.2.8.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.1.2.8.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.8.1.1.2

Add $2$ and $2$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.8.1.2

Multiply $x^{2}$ by $1$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.8.1.3

Multiply $x^{2}$ by $1$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{4} + x^{2} + x^{2} + 1 \cdot 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.8.1.4

Multiply $1$ by $1$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{4} + x^{2} + x^{2} + 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.8.2

Add $x^{2}$ and $x^{2}$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) (x^{4} + 2 x^{2} + 1) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.9

Apply the distributive property.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) x^{4} - \sec^{2} (x - y) (2 x^{2}) - \sec^{2} (x - y) \cdot 1 + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.10

Simplify.

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Step 5.3.1.2.10.1

Rewrite using the commutative property of multiplication.

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) x^{4} - 1 \cdot 2 \sec^{2} (x - y) x^{2} - \sec^{2} (x - y) \cdot 1 + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.10.2

Multiply $- 1$ by $1$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) x^{4} - 1 \cdot 2 \sec^{2} (x - y) x^{2} - \sec^{2} (x - y) + 2 x' x y = x^{2} + 1$

Step 5.3.1.2.11

Multiply $- 1$ by $2$ .

$x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) x^{4} - 2 \sec^{2} (x - y) x^{2} - \sec^{2} (x - y) + 2 x' x y = x^{2} + 1$

Step 5.3.1.3

Reorder factors in $x' \sec^{2} (x - y) x^{4} + 2 x' \sec^{2} (x - y) x^{2} + x' \sec^{2} (x - y) - \sec^{2} (x - y) x^{4} - 2 \sec^{2} (x - y) x^{2} - \sec^{2} (x - y) + 2 x' x y$ .

$x' x^{4} \sec^{2} (x - y) + 2 x' x^{2} \sec^{2} (x - y) + x' \sec^{2} (x - y) - x^{4} \sec^{2} (x - y) - 2 x^{2} \sec^{2} (x - y) - \sec^{2} (x - y) + 2 x' x y = x^{2} + 1$

Step 5.3.2

Move all terms not containing $x'$ to the right side of the equation.

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Step 5.3.2.1

Add $x^{4} \sec^{2} (x - y)$ to both sides of the equation.

$x' x^{4} \sec^{2} (x - y) + 2 x' x^{2} \sec^{2} (x - y) + x' \sec^{2} (x - y) - 2 x^{2} \sec^{2} (x - y) - \sec^{2} (x - y) + 2 x' x y = x^{2} + 1 + x^{4} \sec^{2} (x - y)$

Step 5.3.2.2

Add $2 x^{2} \sec^{2} (x - y)$ to both sides of the equation.

$x' x^{4} \sec^{2} (x - y) + 2 x' x^{2} \sec^{2} (x - y) + x' \sec^{2} (x - y) - \sec^{2} (x - y) + 2 x' x y = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y)$

Step 5.3.2.3

Add $\sec^{2} (x - y)$ to both sides of the equation.

$x' x^{4} \sec^{2} (x - y) + 2 x' x^{2} \sec^{2} (x - y) + x' \sec^{2} (x - y) + 2 x' x y = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3

Factor $x'$ out of $x' x^{4} \sec^{2} (x - y) + 2 x' x^{2} \sec^{2} (x - y) + x' \sec^{2} (x - y) + 2 x' x y$ .

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Step 5.3.3.1

Factor $x'$ out of $x' x^{4} \sec^{2} (x - y)$ .

$x' (x^{4} \sec^{2} (x - y)) + 2 x' x^{2} \sec^{2} (x - y) + x' (\sec^{2} (x - y)) + 2 x' x y = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3.2

Factor $x'$ out of $2 x' x^{2} \sec^{2} (x - y)$ .

$x' (x^{4} \sec^{2} (x - y)) + x' (2 x^{2} \sec^{2} (x - y)) + x' (\sec^{2} (x - y)) + 2 x' x y = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3.3

Factor $x'$ out of $x' \sec^{2} (x - y)$ .

$x' (x^{4} \sec^{2} (x - y)) + x' (2 x^{2} \sec^{2} (x - y)) + x' (\sec^{2} (x - y)) + 2 x' x y = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3.4

Factor $x'$ out of $2 x' x y$ .

$x' (x^{4} \sec^{2} (x - y)) + x' (2 x^{2} \sec^{2} (x - y)) + x' (\sec^{2} (x - y)) + x' (2 x y) = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3.5

Factor $x'$ out of $x' (x^{4} \sec^{2} (x - y)) + x' (2 x^{2} \sec^{2} (x - y))$ .

$x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y)) + x' (\sec^{2} (x - y)) + x' (2 x y) = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3.6

Factor $x'$ out of $x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y)) + x' (\sec^{2} (x - y))$ .

$x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)) + x' (2 x y) = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.3.7

Factor $x'$ out of $x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)) + x' (2 x y)$ .

$x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y) = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$

Step 5.3.4

Divide each term in $x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y) = x^{2} + 1 + x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)$ by $x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y$ and simplify.

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Step 5.3.4.1

$\frac{x' (x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} = \frac{x^{2}}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{x^{4} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

Step 5.3.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{x^{2}}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{x^{4} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Simplify terms.

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Step 5.3.4.3.1.1

Combine the numerators over the common denominator.

$x' = \frac{x^{2} + x^{4} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.1.2

Factor $x^{2}$ out of $x^{2} + x^{4} \sec^{2} (x - y)$ .

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Step 5.3.4.3.1.2.1

Multiply by $1$ .

$x' = \frac{x^{2} \cdot 1 + x^{4} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.1.2.2

Factor $x^{2}$ out of $x^{4} \sec^{2} (x - y)$ .

$x' = \frac{x^{2} \cdot 1 + x^{2} (x^{2} \sec^{2} (x - y))}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.1.2.3

Factor $x^{2}$ out of $x^{2} \cdot 1 + x^{2} (x^{2} \sec^{2} (x - y))$ .

$x' = \frac{x^{2} (1 + x^{2} \sec^{2} (x - y))}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.1.3

Combine the numerators over the common denominator.

$x' = \frac{x^{2} (1 + x^{2} \sec^{2} (x - y)) + 1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.2

Simplify the numerator.

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Step 5.3.4.3.2.1

Apply the distributive property.

$x' = \frac{x^{2} \cdot 1 + x^{2} (x^{2} \sec^{2} (x - y)) + 1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.2.2

Multiply $x^{2}$ by $1$ .

$x' = \frac{x^{2} + x^{2} (x^{2} \sec^{2} (x - y)) + 1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.2.3

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.4.3.2.3.1

Move $x^{2}$ .

$x' = \frac{x^{2} + x^{2} x^{2} \sec^{2} (x - y) + 1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x' = \frac{x^{2} + x^{2 + 2} \sec^{2} (x - y) + 1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.2.3.3

Add $2$ and $2$ .

$x' = \frac{x^{2} + x^{4} \sec^{2} (x - y) + 1}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.3

Combine into one fraction.

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Step 5.3.4.3.3.1

Combine the numerators over the common denominator.

$x' = \frac{x^{2} + x^{4} \sec^{2} (x - y) + 1 + 2 x^{2} \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y} + \frac{\sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 5.3.4.3.3.2

Combine the numerators over the common denominator.

$x' = \frac{x^{2} + x^{4} \sec^{2} (x - y) + 1 + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = \frac{x^{2} + x^{4} \sec^{2} (x - y) + 1 + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 2 x^{2} \sec^{2} (x - y) + \sec^{2} (x - y) + 2 x y}$