Calculus Examples

$2 \cos^{3} (e^{2 x})$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \cos^{3} (e^{2 x})$ with respect to $x$ is $2 \frac{d}{d x} [\cos^{3} (e^{2 x})]$ .

$2 \frac{d}{d x} [\cos^{3} (e^{2 x})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (e^{2 x})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (e^{2 x})$ .

$2 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\cos (e^{2 x})])$

Step 2.2

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

$2 (3 {u_{1}}^{2} \frac{d}{d x} [\cos (e^{2 x})])$

Step 2.3

Replace all occurrences of $u_{1}$ with $\cos (e^{2 x})$ .

$2 (3 \cos^{2} (e^{2 x}) \frac{d}{d x} [\cos (e^{2 x})])$

Step 3

Multiply $3$ by $2$ .

$6 (\cos^{2} (e^{2 x}) \frac{d}{d x} [\cos (e^{2 x})])$

Step 4

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 $g (x) = e^{2 x}$ .

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Step 4.1

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

$6 \cos^{2} (e^{2 x}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [e^{2 x}])$

Step 4.2

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

$6 \cos^{2} (e^{2 x}) (- \sin (u_{2}) \frac{d}{d x} [e^{2 x}])$

Step 4.3

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

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

Step 5

Multiply $- 1$ by $6$ .

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

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 6.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

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

Step 6.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

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

Step 6.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 7

Differentiate.

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Step 7.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 7.2

Multiply $2$ by $- 6$ .

$- 12 \cos^{2} (e^{2 x}) \sin (e^{2 x}) (e^{2 x} (\frac{d}{d x} [x]))$

Step 7.3

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

$- 12 \cos^{2} (e^{2 x}) \sin (e^{2 x}) (e^{2 x} \cdot 1)$

Step 7.4

Simplify the expression.

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Step 7.4.1

Multiply $e^{2 x}$ by $1$ .

$- 12 \cos^{2} (e^{2 x}) \sin (e^{2 x}) e^{2 x}$

Step 7.4.2

Reorder the factors of $- 12 \cos^{2} (e^{2 x}) \sin (e^{2 x}) e^{2 x}$ .

$- 12 e^{2 x} \cos^{2} (e^{2 x}) \sin (e^{2 x})$