Calculus Examples

$x^{\frac{1}{\ln (x)}}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite $x^{\frac{1}{\ln (x)}}$ as $e^{\ln (x^{\frac{1}{\ln (x)}})}$ .

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

Step 1.2

Expand $\ln (x^{\frac{1}{\ln (x)}})$ by moving $\frac{1}{\ln (x)}$ outside the logarithm.

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

Step 2

Simplify terms.

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Step 2.1

Combine $\frac{1}{\ln (x)}$ and $\ln (x)$ .

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

Step 2.2

Cancel the common factor of $\ln (x)$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Simplify.

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

Step 4

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

$0$