Calculus Examples

$y = \cos (4 e^{2 x})$

Step 1

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 $g (x) = 4 e^{2 x}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $4 e^{2 x}$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [4 e^{2 x}]$

Step 1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d x} [4 e^{2 x}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $4 e^{2 x}$ .

$- \sin (4 e^{2 x}) \frac{d}{d x} [4 e^{2 x}]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 e^{2 x}$ with respect to $x$ is $4 \frac{d}{d x} [e^{2 x}]$ .

$- \sin (4 e^{2 x}) (4 \frac{d}{d x} [e^{2 x}])$

Step 2.2

Multiply $4$ by $- 1$ .

$- 4 \sin (4 e^{2 x}) \frac{d}{d x} [e^{2 x}]$

Step 3

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

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Step 3.1

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

$- 4 \sin (4 e^{2 x}) (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [2 x])$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$- 4 \sin (4 e^{2 x}) (e^{u_{2}} \frac{d}{d x} [2 x])$

Step 3.3

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

$- 4 \sin (4 e^{2 x}) (e^{2 x} \frac{d}{d x} [2 x])$

Step 4

Differentiate.

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Step 4.1

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

$- 4 \sin (4 e^{2 x}) (e^{2 x} (2 \frac{d}{d x} [x]))$

Step 4.2

Multiply $2$ by $- 4$ .

$- 8 \sin (4 e^{2 x}) (e^{2 x} (\frac{d}{d x} [x]))$

Step 4.3

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

$- 8 \sin (4 e^{2 x}) (e^{2 x} \cdot 1)$

Step 4.4

Simplify the expression.

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Step 4.4.1

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

$- 8 \sin (4 e^{2 x}) e^{2 x}$

Step 4.4.2

Reorder the factors of $- 8 \sin (4 e^{2 x}) e^{2 x}$ .

$- 8 e^{2 x} \sin (4 e^{2 x})$