Calculus Examples

$f (x) = 3 \ln (x)$

Step 1

Find the first derivative.

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Since $3$ is constant with respect to $x$ , the derivative of $3 \ln (x)$ with respect to $x$ is $3 \frac{d}{d x} [\ln (x)]$ .

$f' (x) = 3 \frac{d}{d x} (\ln (x))$

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

$f' (x) = 3 (\frac{1}{x})$

Combine $3$ and $\frac{1}{x}$ .

$f' (x) = \frac{3}{x}$

Step 2

Find the second derivative.

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Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

$f'' (x) = 3 \frac{d}{d x} (\frac{1}{x})$

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

$f'' (x) = 3 \frac{d}{d x} (x^{- 1})$

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

$f'' (x) = 3 (- x^{- 2})$

Multiply $- 1$ by $3$ .

$f'' (x) = - 3 x^{- 2}$

Simplify.

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

$f'' (x) = - 3 \frac{1}{x^{2}}$

Combine terms.

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Combine $- 3$ and $\frac{1}{x^{2}}$ .

$f'' (x) = \frac{- 3}{x^{2}}$

Move the negative in front of the fraction.

$f'' (x) = - \frac{3}{x^{2}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{3}{x^{2}}$ .

$- \frac{3}{x^{2}}$