Calculus Examples

$y = x - \ln (x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x - \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \ln (x)]$ .

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

Step 1.2

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

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

Step 2

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (x)]$ .

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

Step 2.2

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

$1 - \frac{1}{x}$

Step 3

Reorder terms.

$- \frac{1}{x} + 1$