Calculus Examples

$f (x) = 2 e^{3 x} + \sin (4 x)$

Step 1

By the Sum Rule, the derivative of $2 e^{3 x} + \sin (4 x)$ with respect to $x$ is $\frac{d}{d x} [2 e^{3 x}] + \frac{d}{d x} [\sin (4 x)]$ .

$\frac{d}{d x} [2 e^{3 x}] + \frac{d}{d x} [\sin (4 x)]$

Step 2

Evaluate $\frac{d}{d x} [2 e^{3 x}]$ .

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Step 2.1

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

$2 \frac{d}{d x} [e^{3 x}] + \frac{d}{d x} [\sin (4 x)]$

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$2 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [3 x]) + \frac{d}{d x} [\sin (4 x)]$

Step 2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$2 (e^{u_{1}} \frac{d}{d x} [3 x]) + \frac{d}{d x} [\sin (4 x)]$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$2 (e^{3 x} \frac{d}{d x} [3 x]) + \frac{d}{d x} [\sin (4 x)]$

Step 2.3

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

$2 (e^{3 x} (3 \frac{d}{d x} [x])) + \frac{d}{d x} [\sin (4 x)]$

Step 2.4

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

$2 (e^{3 x} (3 \cdot 1)) + \frac{d}{d x} [\sin (4 x)]$

Step 2.5

Multiply $3$ by $1$ .

$2 (e^{3 x} \cdot 3) + \frac{d}{d x} [\sin (4 x)]$

Step 2.6

Move $3$ to the left of $e^{3 x}$ .

$2 (3 \cdot e^{3 x}) + \frac{d}{d x} [\sin (4 x)]$

Step 2.7

Multiply $3$ by $2$ .

$6 e^{3 x} + \frac{d}{d x} [\sin (4 x)]$

Step 3

Evaluate $\frac{d}{d x} [\sin (4 x)]$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

$6 e^{3 x} + \cos (4 x) (4 \cdot 1)$

Step 3.4

Multiply $4$ by $1$ .

$6 e^{3 x} + \cos (4 x) \cdot 4$

Step 3.5

Move $4$ to the left of $\cos (4 x)$ .

$6 e^{3 x} + 4 \cos (4 x)$