Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} + x^{2} \cos (y)) = \frac{d}{d x} (y^{3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} + x^{2} \cos (y)$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2} \cos (y)]$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.2

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

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Step 2.2.1

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 $f (x) = x^{2}$ and $g (x) = \cos (y)$ .

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

Step 2.2.2

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 $g (x) = y$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

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

$3 x^{2} + x^{2} (- \sin (y) y') + \cos (y) (2 x)$

Step 2.2.5

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

$3 x^{2} + x^{2} (- \sin (y) y') + 2 \cos (y) x$

Step 2.3

Reorder terms.

$3 x^{2} - x^{2} \sin (y) y' + 2 x \cos (y)$

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 $f (x) = x^{3}$ and $g (x) = y$ .

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Step 3.1.1

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

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

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [y]$

Step 3.1.3

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

$3 y^{2} \frac{d}{d x} [y]$

Step 3.2

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

$3 y^{2} y'$

Step 4

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

$3 x^{2} - x^{2} \sin (y) y' + 2 x \cos (y) = 3 y^{2} y'$

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 $3 x^{2} - x^{2} \sin (y) y' + 2 x \cos (y)$ .

$3 x^{2} - x^{2} y' \sin (y) + 2 x \cos (y) = 3 y^{2} y'$

Step 5.2

Subtract $3 y^{2} y'$ from both sides of the equation.

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

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 $3 x^{2}$ from both sides of the equation.

$- x^{2} y' \sin (y) + 2 x \cos (y) - 3 y^{2} y' = - 3 x^{2}$

Step 5.3.2

Subtract $2 x \cos (y)$ from both sides of the equation.

$- x^{2} y' \sin (y) - 3 y^{2} y' = - 3 x^{2} - 2 x \cos (y)$

Step 5.4

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

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Step 5.4.1

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

$y' (- x^{2} \sin (y)) - 3 y^{2} y' = - 3 x^{2} - 2 x \cos (y)$

Step 5.4.2

Factor $y'$ out of $- 3 y^{2} y'$ .

$y' (- x^{2} \sin (y)) + y' (- 3 y^{2}) = - 3 x^{2} - 2 x \cos (y)$

Step 5.4.3

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

$y' (- x^{2} \sin (y) - 3 y^{2}) = - 3 x^{2} - 2 x \cos (y)$

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Move the negative in front of the fraction.

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

Step 5.5.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{3 x^{2}}{- x^{2} \sin (y) - 3 y^{2}} - \frac{2 x \cos (y)}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2

Simplify terms.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 3 x^{2} - 2 x \cos (y)}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2.2

Factor $x$ out of $- 3 x^{2} - 2 x \cos (y)$ .

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Step 5.5.3.2.2.1

Factor $x$ out of $- 3 x^{2}$ .

$y' = \frac{x (- 3 x) - 2 x \cos (y)}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2.2.2

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

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

Step 5.5.3.2.2.3

Factor $x$ out of $x (- 3 x) + x (- 2 \cos (y))$ .

$y' = \frac{x (- 3 x - 2 \cos (y))}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2.3

Factor $- 1$ out of $- 3 x$ .

$y' = \frac{x (- (3 x) - 2 \cos (y))}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2.4

Factor $- 1$ out of $- 2 \cos (y)$ .

$y' = \frac{x (- (3 x) - (2 \cos (y)))}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2.5

Factor $- 1$ out of $- (3 x) - (2 \cos (y))$ .

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

Step 5.5.3.2.6

Rewrite $- (3 x + 2 \cos (y))$ as $- 1 (3 x + 2 \cos (y))$ .

$y' = \frac{x (- 1 (3 x + 2 \cos (y)))}{- x^{2} \sin (y) - 3 y^{2}}$

Step 5.5.3.2.7

Factor $- 1$ out of $- x^{2} \sin (y)$ .

$y' = \frac{x (- 1 (3 x + 2 \cos (y)))}{- (x^{2} \sin (y)) - 3 y^{2}}$

Step 5.5.3.2.8

Factor $- 1$ out of $- 3 y^{2}$ .

$y' = \frac{x (- 1 (3 x + 2 \cos (y)))}{- (x^{2} \sin (y)) - (3 y^{2})}$

Step 5.5.3.2.9

Factor $- 1$ out of $- (x^{2} \sin (y)) - (3 y^{2})$ .

$y' = \frac{x (- 1 (3 x + 2 \cos (y)))}{- (x^{2} \sin (y) + 3 y^{2})}$

Step 5.5.3.2.10

Rewrite $- (x^{2} \sin (y) + 3 y^{2})$ as $- 1 (x^{2} \sin (y) + 3 y^{2})$ .

$y' = \frac{x (- 1 (3 x + 2 \cos (y)))}{- 1 (x^{2} \sin (y) + 3 y^{2})}$

Step 5.5.3.2.11

Cancel the common factor.

$y' = \frac{x (- 1 (3 x + 2 \cos (y)))}{- 1 (x^{2} \sin (y) + 3 y^{2})}$

Step 5.5.3.2.12

Rewrite the expression.

$y' = \frac{x (3 x + 2 \cos (y))}{x^{2} \sin (y) + 3 y^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{x (3 x + 2 \cos (y))}{x^{2} \sin (y) + 3 y^{2}}$