Calculus Examples

$x^{4} + \sin (y) = x^{3} y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{4} + \sin (y)) = \frac{d}{d x} (x^{3} y^{2})$

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 $x^{4} + \sin (y)$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [\sin (y)]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [\sin (y)]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [\sin (y)]$

Step 2.2

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

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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 $f (x) = \sin (x)$ and $g (x) = y$ .

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Step 2.2.1.1

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

$4 x^{3} + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [y]$

Step 2.2.1.2

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

$4 x^{3} + \cos (u) \frac{d}{d x} [y]$

Step 2.2.1.3

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

$4 x^{3} + \cos (y) \frac{d}{d x} [y]$

Step 2.2.2

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

$4 x^{3} + \cos (y) y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x^{3}$ and $g (x) = y^{2}$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Move $2$ to the left of $x^{3}$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Reorder.

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Step 3.6.1

Move $3$ to the left of $y^{2}$ .

$2 x^{3} y y' + 3 \cdot y^{2} x^{2}$

Step 3.6.2

Reorder terms.

$2 x^{3} y y' + 3 x^{2} y^{2}$

Step 4

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

$4 x^{3} + \cos (y) y' = 2 x^{3} y y' + 3 x^{2} y^{2}$

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 $4 x^{3} + \cos (y) y'$ .

$4 x^{3} + y' \cos (y) = 2 x^{3} y y' + 3 x^{2} y^{2}$

Step 5.2

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

$4 x^{3} + y' \cos (y) - 2 x^{3} y y' = 3 x^{2} y^{2}$

Step 5.3

Subtract $4 x^{3}$ from both sides of the equation.

$y' \cos (y) - 2 x^{3} y y' = 3 x^{2} y^{2} - 4 x^{3}$

Step 5.4

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

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Step 5.4.1

Factor $y'$ out of $y' \cos (y)$ .

$y' (\cos (y)) - 2 x^{3} y y' = 3 x^{2} y^{2} - 4 x^{3}$

Step 5.4.2

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

$y' (\cos (y)) + y' (- 2 x^{3} y) = 3 x^{2} y^{2} - 4 x^{3}$

Step 5.4.3

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

$y' (\cos (y) - 2 x^{3} y) = 3 x^{2} y^{2} - 4 x^{3}$

Step 5.5

Divide each term in $y' (\cos (y) - 2 x^{3} y) = 3 x^{2} y^{2} - 4 x^{3}$ by $\cos (y) - 2 x^{3} y$ and simplify.

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Step 5.5.1

Divide each term in $y' (\cos (y) - 2 x^{3} y) = 3 x^{2} y^{2} - 4 x^{3}$ by $\cos (y) - 2 x^{3} y$ .

$\frac{y' (\cos (y) - 2 x^{3} y)}{\cos (y) - 2 x^{3} y} = \frac{3 x^{2} y^{2}}{\cos (y) - 2 x^{3} y} + \frac{- 4 x^{3}}{\cos (y) - 2 x^{3} y}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (\cos (y) - 2 x^{3} y)}{\cos (y) - 2 x^{3} y} = \frac{3 x^{2} y^{2}}{\cos (y) - 2 x^{3} y} + \frac{- 4 x^{3}}{\cos (y) - 2 x^{3} y}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2} y^{2}}{\cos (y) - 2 x^{3} y} + \frac{- 4 x^{3}}{\cos (y) - 2 x^{3} y}$

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.

$y' = \frac{3 x^{2} y^{2}}{\cos (y) - 2 x^{3} y} - \frac{4 x^{3}}{\cos (y) - 2 x^{3} y}$

Step 5.5.3.2

Combine the numerators over the common denominator.

$y' = \frac{3 x^{2} y^{2} - 4 x^{3}}{\cos (y) - 2 x^{3} y}$

Step 5.5.3.3

Factor $x^{2}$ out of $3 x^{2} y^{2} - 4 x^{3}$ .

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Step 5.5.3.3.1

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

$y' = \frac{x^{2} (3 y^{2}) - 4 x^{3}}{\cos (y) - 2 x^{3} y}$

Step 5.5.3.3.2

Factor $x^{2}$ out of $- 4 x^{3}$ .

$y' = \frac{x^{2} (3 y^{2}) + x^{2} (- 4 x)}{\cos (y) - 2 x^{3} y}$

Step 5.5.3.3.3

Factor $x^{2}$ out of $x^{2} (3 y^{2}) + x^{2} (- 4 x)$ .

$y' = \frac{x^{2} (3 y^{2} - 4 x)}{\cos (y) - 2 x^{3} y}$

Step 6

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

$\frac{d y}{d x} = \frac{x^{2} (3 y^{2} - 4 x)}{\cos (y) - 2 x^{3} y}$