Calculus Examples

e^{e x}

$e^{e x}$

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

$g (x) = e x$ .

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Step 1.1

To apply the Chain Rule, set

$u$ as

$e x$ .

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

Step 1.2

Differentiate using the Exponential Rule which states that

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

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

$a$ =

$e$ .

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

Step 1.3

Replace all occurrences of

$u$ with

$e x$ .

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

Step 2

Since

$e$ is constant with respect to

$x$ , the derivative of

$e x$ with respect to

$x$ is

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

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

Step 3

Multiply

$e^{e x}$ by

$e$ by adding the exponents.

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Step 3.1

Move

$e$ .

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

Step 3.2

Multiply

$e$ by

$e^{e x}$ .

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Step 3.2.1

Raise

$e$ to the power of

$1$ .

$e^{1} e^{e x} \frac{d}{d x} [x]$

Step 3.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{1 + e x} \frac{d}{d x} [x]$

Step 4

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$e^{1 + e x} \cdot 1$

Step 5

Multiply

$e^{1 + e x}$ by

$1$ .

$e^{1 + e x}$

$e^{e x}$

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