Calculus Examples

tan (x y) = x

$\tan (x y) = x$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (tan (x y)) = \frac{d}{d x} (x)

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

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 x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \tan (x)$ and

$g (x) = x y$ .

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Step 2.1.1

To apply the Chain Rule, set

$u$ as

$x y$ .

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

Step 2.1.2

The derivative of

$\tan (u)$ with respect to

$u$ is

$\sec^{2} (u)$ .

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

Step 2.1.3

Replace all occurrences of

$u$ with

$x y$ .

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

Step 2.2

Differentiate using the Product Rule which states that

$\frac{d}{d x} [f (x) g (x)]$ is

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

$f (x) = x$ and

$g (x) = y$ .

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

Step 2.3

Rewrite

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

$y'$ .

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

Step 2.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 2.5

Multiply

$y$ by

$1$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Reorder terms.

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

Step 3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$1$

Step 4

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

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

Step 5

Solve for

$y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in

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

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

Step 5.2

Subtract

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

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

Step 5.3

Divide each term in

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

$x \sec^{2} (x y)$ and simplify.

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Step 5.3.1

Divide each term in

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

$x \sec^{2} (x y)$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of

$x$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of

$\sec^{2} (x y)$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide

$y'$ by

$1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of

$\sec^{2} (x y)$ .

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Step 5.3.3.1.1.1

Cancel the common factor.

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

Step 5.3.3.1.1.2

Rewrite the expression.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 6

Replace

$y'$ with

$\frac{d y}{d x}$ .

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