Calculus Examples

$- \frac{6}{5} x^{4} \cos (x)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $x^{4}$ and $\frac{6}{5}$ .

$\frac{d}{d x} [- \frac{x^{4} \cdot 6}{5} \cos (x)]$

Step 1.2

Combine fractions.

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Step 1.2.1

Combine $\cos (x)$ and $\frac{x^{4} \cdot 6}{5}$ .

$\frac{d}{d x} [- \frac{\cos (x) (x^{4} \cdot 6)}{5}]$

Step 1.2.2

Move $6$ to the left of $x^{4}$ .

$\frac{d}{d x} [- \frac{\cos (x) (6 \cdot x^{4})}{5}]$

Step 1.3

Since $- \frac{6}{5}$ is constant with respect to $x$ , the derivative of $- \frac{\cos (x) (6 x^{4})}{5}$ with respect to $x$ is $- \frac{6}{5} \frac{d}{d x} [\cos (x) (x^{4})]$ .

$- \frac{6}{5} \frac{d}{d x} [\cos (x) (x^{4})]$

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 $f (x) = \cos (x)$ and $g (x) = x^{4}$ .

$- \frac{6}{5} (\cos (x) \frac{d}{d x} [x^{4}] + x^{4} \frac{d}{d x} [\cos (x)])$

Step 3

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

$- \frac{6}{5} (\cos (x) (4 x^{3}) + x^{4} \frac{d}{d x} [\cos (x)])$

Step 4

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

$- \frac{6}{5} (\cos (x) (4 x^{3}) + x^{4} (- \sin (x)))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \frac{6}{5} (\cos (x) (4 x^{3})) - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $4$ by $- 1$ .

$- 4 (\frac{6}{5}) (\cos (x) (x^{3})) - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.2

Combine $- 4$ and $\frac{6}{5}$ .

$\frac{- 4 \cdot 6}{5} (\cos (x) (x^{3})) - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.3

Multiply $- 4$ by $6$ .

$\frac{- 24}{5} (\cos (x) (x^{3})) - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.4

Combine $\cos (x)$ and $\frac{- 24}{5}$ .

$\frac{\cos (x) \cdot - 24}{5} x^{3} - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.5

Combine $\frac{\cos (x) \cdot - 24}{5}$ and $x^{3}$ .

$\frac{\cos (x) \cdot - 24 x^{3}}{5} - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.6

Move $- 24$ to the left of $\cos (x)$ .

$\frac{- 24 \cdot \cos (x) x^{3}}{5} - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.7

Move the negative in front of the fraction.

$- \frac{24 \cos (x) x^{3}}{5} - \frac{6}{5} (x^{4} (- \sin (x)))$

Step 5.2.8

Multiply $- 1$ by $- 1$ .

$- \frac{24 \cos (x) x^{3}}{5} + 1 (\frac{6}{5}) (x^{4} (\sin (x)))$

Step 5.2.9

Multiply $\frac{6}{5}$ by $1$ .

$- \frac{24 \cos (x) x^{3}}{5} + \frac{6}{5} (x^{4} (\sin (x)))$

Step 5.2.10

Combine $x^{4}$ and $\frac{6}{5}$ .

$- \frac{24 \cos (x) x^{3}}{5} + \frac{x^{4} \cdot 6}{5} \sin (x)$

Step 5.2.11

Combine $\frac{x^{4} \cdot 6}{5}$ and $\sin (x)$ .

$- \frac{24 \cos (x) x^{3}}{5} + \frac{x^{4} \cdot 6 \sin (x)}{5}$

Step 5.2.12

Move $6$ to the left of $x^{4}$ .

$- \frac{24 \cos (x) x^{3}}{5} + \frac{6 x^{4} \sin (x)}{5}$

Step 5.3

Reorder terms.

$\frac{6 x^{4} \sin (x)}{5} - \frac{24 \cos (x) x^{3}}{5}$

Step 5.4

Reorder factors in $\frac{6 x^{4} \sin (x)}{5} - \frac{24 \cos (x) x^{3}}{5}$ .

$\frac{6 x^{4} \sin (x)}{5} - \frac{24 x^{3} \cos (x)}{5}$