Calculus Examples

$\ln (e^{e^{5 x^{4}}})$

Step 1

Use logarithm rules to move $e^{5 x^{4}}$ out of the exponent.

$\frac{d}{d x} [e^{5 x^{4}} \ln (e)]$

Step 2

The natural logarithm of $e$ is $1$ .

$\frac{d}{d x} [e^{5 x^{4}} \cdot 1]$

Step 3

Multiply $e^{5 x^{4}}$ by $1$ .

$\frac{d}{d x} [e^{5 x^{4}}]$

Step 4

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 $g (x) = 5 x^{4}$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $5 x^{4}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x^{4}]$

Step 4.2

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

$e^{u} \frac{d}{d x} [5 x^{4}]$

Step 4.3

Replace all occurrences of $u$ with $5 x^{4}$ .

$e^{5 x^{4}} \frac{d}{d x} [5 x^{4}]$

Step 5

Differentiate.

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Step 5.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$e^{5 x^{4}} (5 \frac{d}{d x} [x^{4}])$

Step 5.2

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

$e^{5 x^{4}} (5 (4 x^{3}))$

Step 5.3

Multiply $4$ by $5$ .

$e^{5 x^{4}} (20 x^{3})$

Step 6

Simplify.

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Step 6.1

Reorder the factors of $e^{5 x^{4}} (20 x^{3})$ .

$20 e^{5 x^{4}} x^{3}$

Step 6.2

Reorder factors in $20 e^{5 x^{4}} x^{3}$ .

$20 x^{3} e^{5 x^{4}}$