Calculus Examples

$\cos (x y) + x^{7} = y^{7}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cos (x y) + x^{7}) = \frac{d}{d x} (y^{7})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $\cos (x y) + x^{7}$ with respect to $x$ is $\frac{d}{d x} [\cos (x y)] + \frac{d}{d x} [x^{7}]$ .

$\frac{d}{d x} [\cos (x y)] + \frac{d}{d x} [x^{7}]$

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [x y] + \frac{d}{d x} [x^{7}]$

Step 2.2.1.2

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

$- \sin (u) \frac{d}{d x} [x y] + \frac{d}{d x} [x^{7}]$

Step 2.2.1.3

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

$- \sin (x y) \frac{d}{d x} [x y] + \frac{d}{d x} [x^{7}]$

Step 2.2.2

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 $g (x) = y$ .

$- \sin (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [x^{7}]$

Step 2.2.3

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

$- \sin (x y) (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [x^{7}]$

Step 2.2.4

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

$- \sin (x y) (x y' + y \cdot 1) + \frac{d}{d x} [x^{7}]$

Step 2.2.5

Multiply $y$ by $1$ .

$- \sin (x y) (x y' + y) + \frac{d}{d x} [x^{7}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$- \sin (x y) (x y' + y) + 7 x^{6}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- \sin (x y) (x y') - \sin (x y) y + 7 x^{6}$

Step 2.4.2

Reorder terms.

$7 x^{6} - x \sin (x y) y' - y \sin (x y)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = y$ .

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Step 3.1.1

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

$\frac{d}{d u} [u^{7}] \frac{d}{d x} [y]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$7 u^{6} \frac{d}{d x} [y]$

Step 3.1.3

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

$7 y^{6} \frac{d}{d x} [y]$

Step 3.2

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

$7 y^{6} y'$

Step 4

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

$7 x^{6} - x \sin (x y) y' - y \sin (x y) = 7 y^{6} y'$

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 $7 x^{6} - x \sin (x y) y' - y \sin (x y)$ .

$7 x^{6} - x y' \sin (x y) - y \sin (x y) = 7 y^{6} y'$

Step 5.2

Subtract $7 y^{6} y'$ from both sides of the equation.

$7 x^{6} - x y' \sin (x y) - y \sin (x y) - 7 y^{6} y' = 0$

Step 5.3

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

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Step 5.3.1

Subtract $7 x^{6}$ from both sides of the equation.

$- x y' \sin (x y) - y \sin (x y) - 7 y^{6} y' = - 7 x^{6}$

Step 5.3.2

Add $y \sin (x y)$ to both sides of the equation.

$- x y' \sin (x y) - 7 y^{6} y' = - 7 x^{6} + y \sin (x y)$

Step 5.4

Factor $y'$ out of $- x y' \sin (x y) - 7 y^{6} y'$ .

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Step 5.4.1

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

$y' (- x \sin (x y)) - 7 y^{6} y' = - 7 x^{6} + y \sin (x y)$

Step 5.4.2

Factor $y'$ out of $- 7 y^{6} y'$ .

$y' (- x \sin (x y)) + y' (- 7 y^{6}) = - 7 x^{6} + y \sin (x y)$

Step 5.4.3

Factor $y'$ out of $y' (- x \sin (x y)) + y' (- 7 y^{6})$ .

$y' (- x \sin (x y) - 7 y^{6}) = - 7 x^{6} + y \sin (x y)$

Step 5.5

Divide each term in $y' (- x \sin (x y) - 7 y^{6}) = - 7 x^{6} + y \sin (x y)$ by $- x \sin (x y) - 7 y^{6}$ and simplify.

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Step 5.5.1

Divide each term in $y' (- x \sin (x y) - 7 y^{6}) = - 7 x^{6} + y \sin (x y)$ by $- x \sin (x y) - 7 y^{6}$ .

$\frac{y' (- x \sin (x y) - 7 y^{6})}{- x \sin (x y) - 7 y^{6}} = \frac{- 7 x^{6}}{- x \sin (x y) - 7 y^{6}} + \frac{y \sin (x y)}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- x \sin (x y) - 7 y^{6}$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (- x \sin (x y) - 7 y^{6})}{- x \sin (x y) - 7 y^{6}} = \frac{- 7 x^{6}}{- x \sin (x y) - 7 y^{6}} + \frac{y \sin (x y)}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 7 x^{6}}{- x \sin (x y) - 7 y^{6}} + \frac{y \sin (x y)}{- x \sin (x y) - 7 y^{6}}$

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.

$y' = - \frac{7 x^{6}}{- x \sin (x y) - 7 y^{6}} + \frac{y \sin (x y)}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.3.2

Combine the numerators over the common denominator.

$y' = \frac{- 7 x^{6} + y \sin (x y)}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.3.3

Factor $- 1$ out of $- 7 x^{6}$ .

$y' = \frac{- (7 x^{6}) + y \sin (x y)}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.3.4

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

$y' = \frac{- (7 x^{6}) - (- y \sin (x y))}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.3.5

Factor $- 1$ out of $- (7 x^{6}) - (- y \sin (x y))$ .

$y' = \frac{- (7 x^{6} - y \sin (x y))}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.3.6

Rewrite $- (7 x^{6} - y \sin (x y))$ as $- 1 (7 x^{6} - y \sin (x y))$ .

$y' = \frac{- 1 (7 x^{6} - y \sin (x y))}{- x \sin (x y) - 7 y^{6}}$

Step 5.5.3.7

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

$y' = \frac{- 1 (7 x^{6} - y \sin (x y))}{- (x \sin (x y)) - 7 y^{6}}$

Step 5.5.3.8

Factor $- 1$ out of $- 7 y^{6}$ .

$y' = \frac{- 1 (7 x^{6} - y \sin (x y))}{- (x \sin (x y)) - (7 y^{6})}$

Step 5.5.3.9

Factor $- 1$ out of $- (x \sin (x y)) - (7 y^{6})$ .

$y' = \frac{- 1 (7 x^{6} - y \sin (x y))}{- (x \sin (x y) + 7 y^{6})}$

Step 5.5.3.10

Rewrite $- (x \sin (x y) + 7 y^{6})$ as $- 1 (x \sin (x y) + 7 y^{6})$ .

$y' = \frac{- 1 (7 x^{6} - y \sin (x y))}{- 1 (x \sin (x y) + 7 y^{6})}$

Step 5.5.3.11

Cancel the common factor.

$y' = \frac{- 1 (7 x^{6} - y \sin (x y))}{- 1 (x \sin (x y) + 7 y^{6})}$

Step 5.5.3.12

Rewrite the expression.

$y' = \frac{7 x^{6} - y \sin (x y)}{x \sin (x y) + 7 y^{6}}$

Step 6

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

$\frac{d y}{d x} = \frac{7 x^{6} - y \sin (x y)}{x \sin (x y) + 7 y^{6}}$