Calculus Examples

$g (x) = x f^{- 1} (x)$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Combine $\frac{x}{f}$ and $x$ .

$\frac{d}{d x} [\frac{x \cdot x}{f}]$

Step 1.4

Raise $x$ to the power of $1$ .

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

Step 1.5

Raise $x$ to the power of $1$ .

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

Step 1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [\frac{x^{1 + 1}}{f}]$

Step 1.7

Add $1$ and $1$ .

$\frac{d}{d x} [\frac{x^{2}}{f}]$

Step 1.8

Since $\frac{1}{f}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{f}$ with respect to $x$ is $\frac{1}{f} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{f} \frac{d}{d x} [x^{2}]$

Step 1.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{f} (2 x)$

Step 1.10

Combine fractions.

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Step 1.10.1

Combine $2$ and $\frac{1}{f}$ .

$\frac{2}{f} x$

Step 1.10.2

Combine $\frac{2}{f}$ and $x$ .

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

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{2}{f}$ is constant with respect to $x$ , the derivative of $\frac{2 x}{f}$ with respect to $x$ is $\frac{2}{f} \frac{d}{d x} [x]$ .

$\frac{2}{f} \frac{d}{d x} [x]$

Step 2.2

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

$\frac{2}{f} \cdot 1$

Step 2.3

Multiply $\frac{2}{f}$ by $1$ .

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

Step 3

Since $\frac{2}{f}$ is constant with respect to $x$ , the derivative of $\frac{2}{f}$ with respect to $x$ is $0$ .

$f''' (x) = 0$

Step 4

The third derivative of $g (x)$ with respect to $x$ is $0$ .

$0$