Calculus Examples

$x^{e^{x}}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite $x^{e^{x}}$ as $e^{\ln (x^{e^{x}})}$ .

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

Step 1.2

Expand $\ln (x^{e^{x}})$ by moving $e^{x}$ outside the logarithm.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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 $f (x) = e^{x}$ and $g (x) = \ln (x)$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Combine terms.

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Step 7.2.1

Combine $e^{e^{x} \ln (x)}$ and $\frac{e^{x}}{x}$ .

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

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2.3

Multiply $e^{e^{x} \ln (x)}$ by $e^{x}$ by adding the exponents.

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Step 7.2.3.1

Move $e^{x}$ .

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

Step 7.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2.4

To write $e^{x + e^{x} \ln (x)} \ln (x)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 7.2.5

Combine the numerators over the common denominator.

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

Step 7.3

Reorder terms.

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