Calculus Examples

$f (x) = 5 \ln (x)$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

$5 \frac{1}{x}$

Step 1.3

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

$f' (x) = \frac{5}{x}$

Step 2

Find the second derivative.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x}]$ .

$5 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$5 \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$5 (- x^{- 2})$

Step 2.4

Multiply $- 1$ by $5$ .

$- 5 x^{- 2}$

Step 2.5

Simplify.

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Step 2.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 \frac{1}{x^{2}}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 5$ and $\frac{1}{x^{2}}$ .

$\frac{- 5}{x^{2}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{5}{x^{2}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{5}{x^{2}}$ .

$- \frac{5}{x^{2}}$