Calculus Examples

$\frac{f (3 + h) - f \cdot 3}{h}$

Step 1

Multiply $3$ by $- 1$ .

$\frac{d}{d h} [\frac{f (3 + h) - 3 f}{h}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d h} [\frac{f (h)}{g (h)}]$ is $\frac{g (h) \frac{d}{d h} [f (h)] - f (h) \frac{d}{d h} [g (h)]}{{g (h)}^{2}}$ where $f (h) = f (3 + h) - 3 f$ and $g (h) = h$ .

$\frac{h \frac{d}{d h} [f (3 + h) - 3 f] - (f (3 + h) - 3 f) \frac{d}{d h} [h]}{h^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $f (3 + h) - 3 f$ with respect to $h$ is $\frac{d}{d h} [f (3 + h)] + \frac{d}{d h} [- 3 f]$ .

$\frac{h (\frac{d}{d h} [f (3 + h)] + \frac{d}{d h} [- 3 f]) - (f (3 + h) - 3 f) \frac{d}{d h} [h]}{h^{2}}$

Step 3.2

Since $- 3 f$ is constant with respect to $h$ , the derivative of $- 3 f$ with respect to $h$ is $0$ .

$\frac{h (\frac{d}{d h} [f (3 + h)] + 0) - (f (3 + h) - 3 f) \frac{d}{d h} [h]}{h^{2}}$

Step 3.3

Add $\frac{d}{d h} [f (3 + h)]$ and $0$ .

$\frac{h \frac{d}{d h} [f (3 + h)] - (f (3 + h) - 3 f) \frac{d}{d h} [h]}{h^{2}}$

Step 3.4

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

$\frac{h \frac{d}{d h} [f (3 + h)] - (f (3 + h) - 3 f) \cdot 1}{h^{2}}$

Step 3.5

Multiply $- 1$ by $1$ .

$\frac{h \frac{d}{d h} [f (3 + h)] - (f (3 + h) - 3 f)}{h^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{h \frac{d}{d h} [f (3 + h)] - f (3 + h) - (- 3 f)}{h^{2}}$

Step 4.2

Multiply $- 3$ by $- 1$ .

$\frac{h \frac{d}{d h} [f (3 + h)] - f (3 + h) + 3 f}{h^{2}}$

Step 4.3

Reorder terms.

$\frac{h \frac{d}{d h} [f (3 + h)] + 3 f - f (3 + h)}{h^{2}}$