Calculus Examples

$y = e^{x^{5}} \ln (x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x^{5}}$ and $g (x) = \ln (x)$ .

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

Step 2

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

$e^{x^{5}} \frac{1}{x} + \ln (x) \frac{d}{d x} [e^{x^{5}}]$

Step 3

Combine $e^{x^{5}}$ and $\frac{1}{x}$ .

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

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

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

Step 6

Simplify.

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Step 6.1

Reorder terms.

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

Step 6.2

Reorder factors in $5 e^{x^{5}} x^{4} \ln (x) + \frac{e^{x^{5}}}{x}$ .

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