Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{3} y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x^{3} y$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $x^{3} y$ .

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

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Move $3$ to the left of $y$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Reorder terms.

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

Step 3

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

$3 x^{2}$

Step 4

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

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

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

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

Step 5.2

Subtract $3 x^{2} y \cos (x^{3} y)$ from both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x^{3}$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

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

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

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 $x^{2}$ and $x^{3}$ .

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Step 5.3.3.1.1.1

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

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

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

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

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

Step 5.3.3.1.1.2.2

Cancel the common factor.

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

Step 5.3.3.1.1.2.3

Rewrite the expression.

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

Step 5.3.3.1.2

Separate fractions.

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

Step 5.3.3.1.3

Convert from $\frac{1}{\cos (x^{3} y)}$ to $\sec (x^{3} y)$ .

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

Step 5.3.3.1.4

Combine $\frac{3}{x}$ and $\sec (x^{3} y)$ .

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

Step 5.3.3.1.5

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.3.3.1.5.1

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

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

Step 5.3.3.1.5.2

Cancel the common factors.

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Step 5.3.3.1.5.2.1

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

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

Step 5.3.3.1.5.2.2

Cancel the common factor.

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

Step 5.3.3.1.5.2.3

Rewrite the expression.

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

Step 5.3.3.1.6

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

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Step 5.3.3.1.6.1

Cancel the common factor.

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

Step 5.3.3.1.6.2

Rewrite the expression.

$y' = \frac{3 \sec (x^{3} y)}{x} + \frac{- 3 y}{x}$

Step 5.3.3.1.7

Move the negative in front of the fraction.

$y' = \frac{3 \sec (x^{3} y)}{x} - \frac{3 y}{x}$

Step 6

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

$\frac{d y}{d x} = \frac{3 \sec (x^{3} y)}{x} - \frac{3 y}{x}$