Calculus Examples

$\ln^{4} (1 + e^{x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \ln (1 + e^{x})$ .

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Step 1.1

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

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

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

Step 1.3

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 1 + e^{x}$ .

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Step 2.1

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

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

Step 2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$ .

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

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 \ln^{3} (1 + e^{x}) (\frac{1}{1 + e^{x}} (0 + \frac{d}{d x} [e^{x}]))$

Step 3.3

Add $0$ and $\frac{d}{d x} [e^{x}]$ .

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

Step 4

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

$4 \ln^{3} (1 + e^{x}) (\frac{1}{1 + e^{x}} e^{x})$

Step 5

Combine $\frac{1}{1 + e^{x}}$ and $e^{x}$ .

$4 \ln^{3} (1 + e^{x}) \frac{e^{x}}{1 + e^{x}}$

Step 6

Combine terms.

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Step 6.1

Combine $\frac{e^{x}}{1 + e^{x}}$ and $4$ .

$\frac{e^{x} \cdot 4}{1 + e^{x}} \ln^{3} (1 + e^{x})$

Step 6.2

Combine $\frac{e^{x} \cdot 4}{1 + e^{x}}$ and $\ln^{3} (1 + e^{x})$ .

$\frac{e^{x} \cdot 4 \ln^{3} (1 + e^{x})}{1 + e^{x}}$

Step 6.3

Move $4$ to the left of $e^{x}$ .

$\frac{4 e^{x} \ln^{3} (1 + e^{x})}{1 + e^{x}}$