Calculus Examples

$\tan (3 x + y) = 3 x$ , $(0, 0)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\tan (3 x + y)) = \frac{d}{d y} (3 x)$

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 $g (y) = 3 x + y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $3 x + y$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d y} [3 x + y]$

Step 2.1.2

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

$\sec^{2} (u) \frac{d}{d y} [3 x + y]$

Step 2.1.3

Replace all occurrences of $u$ with $3 x + y$ .

$\sec^{2} (3 x + y) \frac{d}{d y} [3 x + y]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $3 x + y$ with respect to $y$ is $\frac{d}{d y} [3 x] + \frac{d}{d y} [y]$ .

$\sec^{2} (3 x + y) (\frac{d}{d y} [3 x] + \frac{d}{d y} [y])$

Step 2.2.2

Since $3$ is constant with respect to $y$ , the derivative of $3 x$ with respect to $y$ is $3 \frac{d}{d y} [x]$ .

$\sec^{2} (3 x + y) (3 \frac{d}{d y} [x] + \frac{d}{d y} [y])$

Step 2.3

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

$\sec^{2} (3 x + y) (3 x' + \frac{d}{d y} [y])$

Step 2.4

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

$\sec^{2} (3 x + y) (3 x' + 1)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $3$ is constant with respect to $y$ , the derivative of $3 x$ with respect to $y$ is $3 \frac{d}{d y} [x]$ .

$3 \frac{d}{d y} [x]$

Step 3.2

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

$3 x'$

Step 4

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

$\sec^{2} (3 x + y) (3 x' + 1) = 3 x'$

Step 5

Solve for $x'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\sec^{2} (3 x + y) (3 x' + 1)$ .

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Step 5.1.1.1

Rewrite.

$0 + 0 + \sec^{2} (3 x + y) (3 x' + 1) = 3 x'$

Step 5.1.1.2

Simplify by adding zeros.

$\sec^{2} (3 x + y) (3 x' + 1) = 3 x'$

Step 5.1.1.3

Apply the distributive property.

$\sec^{2} (3 x + y) (3 x') + \sec^{2} (3 x + y) \cdot 1 = 3 x'$

Step 5.1.1.4

Simplify the expression.

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Step 5.1.1.4.1

Rewrite using the commutative property of multiplication.

$3 \sec^{2} (3 x + y) x' + \sec^{2} (3 x + y) \cdot 1 = 3 x'$

Step 5.1.1.4.2

Multiply $\sec^{2} (3 x + y)$ by $1$ .

$3 \sec^{2} (3 x + y) x' + \sec^{2} (3 x + y) = 3 x'$

Step 5.1.1.4.3

Reorder factors in $3 \sec^{2} (3 x + y) x' + \sec^{2} (3 x + y)$ .

$3 x' \sec^{2} (3 x + y) + \sec^{2} (3 x + y) = 3 x'$

Step 5.2

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

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Step 5.2.1

Subtract $3 x'$ from both sides of the equation.

$3 x' \sec^{2} (3 x + y) + \sec^{2} (3 x + y) - 3 x' = 0$

Step 5.2.2

Move $- 3 x'$ .

$3 x' \sec^{2} (3 x + y) - 3 x' + \sec^{2} (3 x + y) = 0$

Step 5.2.3

Factor $3 x'$ out of $3 x' \sec^{2} (3 x + y)$ .

$3 x' \sec^{2} (3 x + y) - 3 x' + \sec^{2} (3 x + y) = 0$

Step 5.2.4

Factor $3 x'$ out of $- 3 x'$ .

$3 x' \sec^{2} (3 x + y) + 3 x' \cdot - 1 + \sec^{2} (3 x + y) = 0$

Step 5.2.5

Factor $3 x'$ out of $3 x' \sec^{2} (3 x + y) + 3 x' \cdot - 1$ .

$3 x' (\sec^{2} (3 x + y) - 1) + \sec^{2} (3 x + y) = 0$

Step 5.2.6

Apply pythagorean identity.

$3 x' \tan^{2} (3 x + y) + \sec^{2} (3 x + y) = 0$

Step 5.3

Subtract $\sec^{2} (3 x + y)$ from both sides of the equation.

$3 x' \tan^{2} (3 x + y) = - \sec^{2} (3 x + y)$

Step 5.4

Divide each term in $3 x' \tan^{2} (3 x + y) = - \sec^{2} (3 x + y)$ by $3 \tan^{2} (3 x + y)$ and simplify.

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Step 5.4.1

Divide each term in $3 x' \tan^{2} (3 x + y) = - \sec^{2} (3 x + y)$ by $3 \tan^{2} (3 x + y)$ .

$\frac{3 x' \tan^{2} (3 x + y)}{3 \tan^{2} (3 x + y)} = \frac{- \sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{3 x' \tan^{2} (3 x + y)}{3 \tan^{2} (3 x + y)} = \frac{- \sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x' \tan^{2} (3 x + y)}{\tan^{2} (3 x + y)} = \frac{- \sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 5.4.2.2

Cancel the common factor of $\tan^{2} (3 x + y)$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{x' \tan^{2} (3 x + y)}{\tan^{2} (3 x + y)} = \frac{- \sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 5.4.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{- \sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$x' = - \frac{\sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 6

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

$\frac{d x}{d y} = - \frac{\sec^{2} (3 x + y)}{3 \tan^{2} (3 x + y)}$

Step 7

Replace $x$ with $0$ and $y$ with $0$ in the expression.

$\frac{d x}{d y} = - \frac{\sec^{2} (3 (0) + 0)}{3 \tan^{2} (3 (0) + 0)}$

Step 8

Simplify the result.

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Step 8.1

Remove parentheses.

$- \frac{\sec^{2} (3 (0) + 0)}{3 \tan^{2} (3 (0) + 0)}$

Step 8.2

Multiply $3$ by $0$ .

$- \frac{\sec^{2} (3 (0) + 0)}{3 \tan^{2} (0 + 0)}$

Step 8.3

Add $0$ and $0$ .

$- \frac{\sec^{2} (3 (0) + 0)}{3 \tan^{2} (0)}$

Step 8.4

The exact value of $\tan (0)$ is $0$ .

$- \frac{\sec^{2} (3 (0) + 0)}{3 \cdot 0^{2}}$

Step 8.5

Raising $0$ to any positive power yields $0$ .

$- \frac{\sec^{2} (3 (0) + 0)}{3 \cdot 0}$

Step 8.6

Multiply $3$ by $0$ .

$- \frac{\sec^{2} (3 (0) + 0)}{0}$

Step 8.7

The expression contains a division by $0$ . The expression is undefined.

Undefined