Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

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

Step 1.2

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{x^{3} \cos (x^{3})}{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{3} \cos (x^{3})]$ .

$- \frac{1}{3} \frac{d}{d x} [x^{3} \cos (x^{3})]$

Step 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 $g (x) = \cos (x^{3})$ .

$- \frac{1}{3} (x^{3} \frac{d}{d x} [\cos (x^{3})] + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 3

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 $g (x) = x^{3}$ .

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Step 3.1

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

$- \frac{1}{3} (x^{3} (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [x^{3}]) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.2

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

$- \frac{1}{3} (x^{3} (- \sin (u) \frac{d}{d x} [x^{3}]) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.3

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

$- \frac{1}{3} (x^{3} (- \sin (x^{3}) \frac{d}{d x} [x^{3}]) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$- \frac{1}{3} (x^{3} (- \sin (x^{3}) (3 x^{2})) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 4.2

Multiply $3$ by $- 1$ .

$- \frac{1}{3} (x^{3} (- 3 \sin (x^{3}) x^{2}) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 5.1

Move $x^{2}$ .

$- \frac{1}{3} (x^{2} x^{3} (- 3 \sin (x^{3})) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{3} (x^{2 + 3} (- 3 \sin (x^{3})) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 5.3

Add $2$ and $3$ .

$- \frac{1}{3} (x^{5} (- 3 \sin (x^{3})) + \cos (x^{3}) \frac{d}{d x} [x^{3}])$

Step 6

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

$- \frac{1}{3} (x^{5} (- 3 \sin (x^{3})) + \cos (x^{3}) (3 x^{2}))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$- \frac{1}{3} (x^{5} (- 3 \sin (x^{3}))) - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2

Combine terms.

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Step 7.2.1

Multiply $- 3$ by $- 1$ .

$3 (\frac{1}{3}) (x^{5} (\sin (x^{3}))) - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.2

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} (x^{5} (\sin (x^{3}))) - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.3

Combine $x^{5}$ and $\frac{3}{3}$ .

$\frac{x^{5} \cdot 3}{3} \sin (x^{3}) - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.4

Combine $\frac{x^{5} \cdot 3}{3}$ and $\sin (x^{3})$ .

$\frac{x^{5} \cdot 3 \sin (x^{3})}{3} - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.5

Move $3$ to the left of $x^{5}$ .

$\frac{3 \cdot x^{5} \sin (x^{3})}{3} - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.6

Cancel the common factor of $3$ .

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Step 7.2.6.1

Cancel the common factor.

$\frac{3 x^{5} \sin (x^{3})}{3} - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.6.2

Divide $x^{5} \sin (x^{3})$ by $1$ .

$x^{5} \sin (x^{3}) - \frac{1}{3} (\cos (x^{3}) (3 x^{2}))$

Step 7.2.7

Multiply $3$ by $- 1$ .

$x^{5} \sin (x^{3}) - 3 (\frac{1}{3}) (\cos (x^{3}) (x^{2}))$

Step 7.2.8

Combine $- 3$ and $\frac{1}{3}$ .

$x^{5} \sin (x^{3}) + \frac{- 3}{3} (\cos (x^{3}) (x^{2}))$

Step 7.2.9

Combine $\cos (x^{3})$ and $\frac{- 3}{3}$ .

$x^{5} \sin (x^{3}) + \frac{\cos (x^{3}) \cdot - 3}{3} x^{2}$

Step 7.2.10

Combine $\frac{\cos (x^{3}) \cdot - 3}{3}$ and $x^{2}$ .

$x^{5} \sin (x^{3}) + \frac{\cos (x^{3}) \cdot - 3 x^{2}}{3}$

Step 7.2.11

Move $- 3$ to the left of $\cos (x^{3})$ .

$x^{5} \sin (x^{3}) + \frac{- 3 \cdot \cos (x^{3}) x^{2}}{3}$

Step 7.2.12

Cancel the common factor of $- 3$ and $3$ .

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Step 7.2.12.1

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

$x^{5} \sin (x^{3}) + \frac{3 (- \cos (x^{3}) x^{2})}{3}$

Step 7.2.12.2

Cancel the common factors.

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Step 7.2.12.2.1

Factor $3$ out of $3$ .

$x^{5} \sin (x^{3}) + \frac{3 (- \cos (x^{3}) x^{2})}{3 (1)}$

Step 7.2.12.2.2

Cancel the common factor.

$x^{5} \sin (x^{3}) + \frac{3 (- \cos (x^{3}) x^{2})}{3 \cdot 1}$

Step 7.2.12.2.3

Rewrite the expression.

$x^{5} \sin (x^{3}) + \frac{- \cos (x^{3}) x^{2}}{1}$

Step 7.2.12.2.4

Divide $- \cos (x^{3}) x^{2}$ by $1$ .

$x^{5} \sin (x^{3}) - \cos (x^{3}) x^{2}$

Step 7.3

Reorder terms.

$x^{5} \sin (x^{3}) - x^{2} \cos (x^{3})$