Calculus Examples

f (x) = e^{3 x}

$f (x) = e^{3 x}$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = e^{x}$ and

$g (x) = 3 x$ .

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Step 1.1

To apply the Chain Rule, set

$u$ as

$3 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x]$

Step 1.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d u} [a^{u}]$ is

$a^{u} \ln (a)$ where

$a$ =

$e$ .

$e^{u} \frac{d}{d x} [3 x]$

Step 1.3

Replace all occurrences of

$u$ with

$3 x$ .

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

Step 2

Differentiate.

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Step 2.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

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

Step 2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$e^{3 x} (3 \cdot 1)$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply

$3$ by

$1$ .

$e^{3 x} \cdot 3$

Step 2.3.2

Move

$3$ to the left of

$e^{3 x}$ .

$3 e^{3 x}$