Algebra Examples

$x^{2} \cos (y) + \sin (2 y) = x y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x^{2} \cos (y) + \sin (2 y)) = \frac{d}{d y} (x y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

$\frac{d}{d y} [x^{2} \cos (y)] + \frac{d}{d y} [\sin (2 y)]$

Step 2.2

Evaluate $\frac{d}{d y} [x^{2} \cos (y)]$ .

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Step 2.2.1

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

$x^{2} \frac{d}{d y} [\cos (y)] + \cos (y) \frac{d}{d y} [x^{2}] + \frac{d}{d y} [\sin (2 y)]$

Step 2.2.2

The derivative of $\cos (y)$ with respect to $y$ is $- \sin (y)$ .

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

Step 2.2.3

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 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x$ .

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

Step 2.2.4

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

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

Step 2.2.5

Move $2$ to the left of $\cos (y)$ .

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

Step 2.3

Evaluate $\frac{d}{d y} [\sin (2 y)]$ .

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Step 2.3.1

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

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Step 2.3.1.1

To apply the Chain Rule, set $u_{2}$ as $2 y$ .

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

Step 2.3.1.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

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

Step 2.3.1.3

Replace all occurrences of $u_{2}$ with $2 y$ .

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

Step 2.3.2

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

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

Step 2.3.3

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

$x^{2} (- \sin (y)) + 2 \cos (y) x x' + \cos (2 y) (2 \cdot 1)$

Step 2.3.4

Multiply $2$ by $1$ .

$x^{2} (- \sin (y)) + 2 \cos (y) x x' + \cos (2 y) \cdot 2$

Step 2.3.5

Move $2$ to the left of $\cos (2 y)$ .

$x^{2} (- \sin (y)) + 2 \cos (y) x x' + 2 \cos (2 y)$

Step 2.4

Reorder terms.

$- x^{2} \sin (y) + 2 x \cos (y) x' + 2 \cos (2 y)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$x \frac{d}{d y} [y] + y \frac{d}{d y} [x]$

Step 3.2

Differentiate using the Power Rule.

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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 $n = 1$ .

$x \cdot 1 + y \frac{d}{d y} [x]$

Step 3.2.2

Multiply $x$ by $1$ .

$x + y \frac{d}{d y} [x]$

Step 3.3

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

$x + y x'$

Step 3.4

Reorder terms.

$y x' + x$

Step 4

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

$- x^{2} \sin (y) + 2 x \cos (y) x' + 2 \cos (2 y) = y x' + x$

Step 5

Solve for $x'$ .

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Step 5.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Subtract $2 (1 - 2 \sin^{2} (y))$ from both sides of the equation.

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

Step 5.3

Subtract $y x'$ from both sides of the equation.

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

Step 5.4

Simplify the left side.

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Step 5.4.1

Reorder factors in $2 x \cos (y) x' - y x'$ .

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

Step 5.5

Simplify the right side.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Apply the distributive property.

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

Step 5.5.1.2

Multiply $- 2$ by $1$ .

$2 x x' \cos (y) - y x' = x + x^{2} \sin (y) - 2 - 2 (- 2 \sin^{2} (y))$

Step 5.5.1.3

Multiply $- 2$ by $- 2$ .

$2 x x' \cos (y) - y x' = x + x^{2} \sin (y) - 2 + 4 \sin^{2} (y)$

Step 5.6

Solve the equation for $x'$ .

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Step 5.6.1

Factor $x'$ out of $2 x x' \cos (y) - y x'$ .

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Step 5.6.1.1

Factor $x'$ out of $2 x x' \cos (y)$ .

$x' (2 x \cos (y)) - y x' = x + x^{2} \sin (y) - 2 + 4 \sin^{2} (y)$

Step 5.6.1.2

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

$x' (2 x \cos (y)) + x' (- y) = x + x^{2} \sin (y) - 2 + 4 \sin^{2} (y)$

Step 5.6.1.3

Factor $x'$ out of $x' (2 x \cos (y)) + x' (- y)$ .

$x' (2 x \cos (y) - y) = x + x^{2} \sin (y) - 2 + 4 \sin^{2} (y)$

Step 5.6.2

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

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Step 5.6.2.1

Divide each term in $x' (2 x \cos (y) - y) = x + x^{2} \sin (y) - 2 + 4 \sin^{2} (y)$ by $2 x \cos (y) - y$ .

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

Step 5.6.2.2

Simplify the left side.

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Step 5.6.2.2.1

Cancel the common factor of $2 x \cos (y) - y$ .

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Step 5.6.2.2.1.1

Cancel the common factor.

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

Step 5.6.2.2.1.2

Divide $x'$ by $1$ .

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

Step 5.6.2.3

Simplify the right side.

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Step 5.6.2.3.1

Simplify terms.

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Step 5.6.2.3.1.1

Move the negative in front of the fraction.

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

Step 5.6.2.3.1.2

Combine the numerators over the common denominator.

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

Step 5.6.2.3.1.3

Factor $x$ out of $x + x^{2} \sin (y)$ .

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Step 5.6.2.3.1.3.1

Raise $x$ to the power of $1$ .

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

Step 5.6.2.3.1.3.2

Factor $x$ out of $x^{1}$ .

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

Step 5.6.2.3.1.3.3

Factor $x$ out of $x^{2} \sin (y)$ .

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

Step 5.6.2.3.1.3.4

Factor $x$ out of $x \cdot 1 + x (x \sin (y))$ .

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

Step 5.6.2.3.1.4

Combine the numerators over the common denominator.

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

Step 5.6.2.3.2

Simplify the numerator.

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Step 5.6.2.3.2.1

Apply the distributive property.

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

Step 5.6.2.3.2.2

Multiply $x$ by $1$ .

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

Step 5.6.2.3.2.3

Multiply $x$ by $x$ by adding the exponents.

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Step 5.6.2.3.2.3.1

Move $x$ .

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

Step 5.6.2.3.2.3.2

Multiply $x$ by $x$ .

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

Step 5.6.2.3.3

Combine the numerators over the common denominator.

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

Step 6

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

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