Calculus Examples

$f (x) = - 3 \cos (5 x^{3} + 7 x)$

Step 1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \cos (5 x^{3} + 7 x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\cos (5 x^{3} + 7 x)]$ .

$- 3 \frac{d}{d x} [\cos (5 x^{3} + 7 x)]$

Step 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 $g (x) = 5 x^{3} + 7 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $5 x^{3} + 7 x$ .

$- 3 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [5 x^{3} + 7 x])$

Step 2.2

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

$- 3 (- \sin (u) \frac{d}{d x} [5 x^{3} + 7 x])$

Step 2.3

Replace all occurrences of $u$ with $5 x^{3} + 7 x$ .

$- 3 (- \sin (5 x^{3} + 7 x) \frac{d}{d x} [5 x^{3} + 7 x])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $- 3$ .

$3 (\sin (5 x^{3} + 7 x) \frac{d}{d x} [5 x^{3} + 7 x])$

Step 3.2

By the Sum Rule, the derivative of $5 x^{3} + 7 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [7 x]$ .

$3 \sin (5 x^{3} + 7 x) (\frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [7 x])$

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$3 \sin (5 x^{3} + 7 x) (5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7 x])$

Step 3.4

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

$3 \sin (5 x^{3} + 7 x) (5 (3 x^{2}) + \frac{d}{d x} [7 x])$

Step 3.5

Multiply $3$ by $5$ .

$3 \sin (5 x^{3} + 7 x) (15 x^{2} + \frac{d}{d x} [7 x])$

Step 3.6

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$3 \sin (5 x^{3} + 7 x) (15 x^{2} + 7 \frac{d}{d x} [x])$

Step 3.7

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

$3 \sin (5 x^{3} + 7 x) (15 x^{2} + 7 \cdot 1)$

Step 3.8

Multiply $7$ by $1$ .

$3 \sin (5 x^{3} + 7 x) (15 x^{2} + 7)$