Calculus Examples

$\tan (x y) = x$

Step 1

Differentiate both sides of the equation.

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

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 $g (y) = x y$ .

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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

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 $g (y) = y$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $x$ by $1$ .

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

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Reorder terms.

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

Step 3

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

$x'$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $y \sec^{2} (x y) - 1$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $x'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 6

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

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