Calculus Examples

$f (x) = \ln (x^{3})$

Step 1

Find the first derivative.

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Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{3}$ .

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To apply the Chain Rule, set $u$ as $x^{3}$ .

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

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

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

Replace all occurrences of $u$ with $x^{3}$ .

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

Differentiate using the Power Rule.

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Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Simplify terms.

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

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

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

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

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Factor $x^{2}$ out of $3 x^{2}$ .

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

Cancel the common factors.

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Factor $x^{2}$ out of $x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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