Calculus Examples

$f (x) = 7 \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) - 2$

Step 1

By the Sum Rule, the derivative of $7 \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) - 2$ with respect to $x$ is $\frac{d}{d x} [7 \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [7 \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2

Evaluate $\frac{d}{d x} [7 \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x) \cdot \cos (6 x)]$ .

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Step 2.1

Raise $\cos (6 x)$ to the power of $1$ .

$\frac{d}{d x} [7 (\cos^{1} (6 x) \cos (6 x)) \cdot \cos (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.2

Raise $\cos (6 x)$ to the power of $1$ .

$\frac{d}{d x} [7 (\cos^{1} (6 x) \cos^{1} (6 x)) \cdot \cos (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.3

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

$\frac{d}{d x} [7 {\cos (6 x)}^{1 + 1} \cdot \cos (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.4

Add $1$ and $1$ .

$\frac{d}{d x} [7 \cos^{2} (6 x) \cdot \cos (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.5

Raise $\cos (6 x)$ to the power of $1$ .

$\frac{d}{d x} [7 (\cos^{1} (6 x) \cos^{2} (6 x)) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.6

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

$\frac{d}{d x} [7 {\cos (6 x)}^{1 + 2} \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.7

Add $1$ and $2$ .

$\frac{d}{d x} [7 \cos^{3} (6 x) \cdot \cos (6 x)] + \frac{d}{d x} [- 2]$

Step 2.8

Raise $\cos (6 x)$ to the power of $1$ .

$\frac{d}{d x} [7 (\cos^{1} (6 x) \cos^{3} (6 x))] + \frac{d}{d x} [- 2]$

Step 2.9

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

$\frac{d}{d x} [7 {\cos (6 x)}^{1 + 3}] + \frac{d}{d x} [- 2]$

Step 2.10

Add $1$ and $3$ .

$\frac{d}{d x} [7 \cos^{4} (6 x)] + \frac{d}{d x} [- 2]$

Step 2.11

Since $7$ is constant with respect to $x$ , the derivative of $7 \cos^{4} (6 x)$ with respect to $x$ is $7 \frac{d}{d x} [\cos^{4} (6 x)]$ .

$7 \frac{d}{d x} [\cos^{4} (6 x)] + \frac{d}{d x} [- 2]$

Step 2.12

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \cos (6 x)$ .

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Step 2.12.1

To apply the Chain Rule, set $u_{1}$ as $\cos (6 x)$ .

$7 (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\cos (6 x)]) + \frac{d}{d x} [- 2]$

Step 2.12.2

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

$7 (4 {u_{1}}^{3} \frac{d}{d x} [\cos (6 x)]) + \frac{d}{d x} [- 2]$

Step 2.12.3

Replace all occurrences of $u_{1}$ with $\cos (6 x)$ .

$7 (4 \cos^{3} (6 x) \frac{d}{d x} [\cos (6 x)]) + \frac{d}{d x} [- 2]$

Step 2.13

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 $g (x) = 6 x$ .

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Step 2.13.1

To apply the Chain Rule, set $u_{2}$ as $6 x$ .

$7 (4 \cos^{3} (6 x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [6 x])) + \frac{d}{d x} [- 2]$

Step 2.13.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$7 (4 \cos^{3} (6 x) (- \sin (u_{2}) \frac{d}{d x} [6 x])) + \frac{d}{d x} [- 2]$

Step 2.13.3

Replace all occurrences of $u_{2}$ with $6 x$ .

$7 (4 \cos^{3} (6 x) (- \sin (6 x) \frac{d}{d x} [6 x])) + \frac{d}{d x} [- 2]$

Step 2.14

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$7 (4 \cos^{3} (6 x) (- \sin (6 x) (6 \frac{d}{d x} [x]))) + \frac{d}{d x} [- 2]$

Step 2.15

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

$7 (4 \cos^{3} (6 x) (- \sin (6 x) (6 \cdot 1))) + \frac{d}{d x} [- 2]$

Step 2.16

Multiply $6$ by $1$ .

$7 (4 \cos^{3} (6 x) (- \sin (6 x) \cdot 6)) + \frac{d}{d x} [- 2]$

Step 2.17

Multiply $6$ by $- 1$ .

$7 (4 \cos^{3} (6 x) (- 6 \sin (6 x))) + \frac{d}{d x} [- 2]$

Step 2.18

Multiply $- 6$ by $4$ .

$7 (- 24 \cos^{3} (6 x) \sin (6 x)) + \frac{d}{d x} [- 2]$

Step 2.19

Multiply $- 24$ by $7$ .

$- 168 \cos^{3} (6 x) \sin (6 x) + \frac{d}{d x} [- 2]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- 168 \cos^{3} (6 x) \sin (6 x) + 0$

Step 3.2

Add $- 168 \cos^{3} (6 x) \sin (6 x)$ and $0$ .

$- 168 \cos^{3} (6 x) \sin (6 x)$