Calculus Examples

$\frac{f (x + h) - f (x)}{h}$

Step 1

Apply the distributive property.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

$\frac{1}{h} (f \cdot 1 + \frac{d}{d x} [f h] + \frac{d}{d x} [- f (x)])$

Step 6

Multiply $f$ by $1$ .

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

Step 7

Since $f h$ is constant with respect to $x$ , the derivative of $f h$ with respect to $x$ is $0$ .

$\frac{1}{h} (f + 0 + \frac{d}{d x} [- f (x)])$

Step 8

Add $f$ and $0$ .

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

Step 9

Since $- f (x)$ is constant with respect to $x$ , the derivative of $- f (x)$ with respect to $x$ is $0$ .

$\frac{1}{h} (f + 0)$

Step 10

Combine fractions.

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Step 10.1

Add $f$ and $0$ .

$\frac{1}{h} f$

Step 10.2

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

$\frac{f}{h}$