Calculus Examples

$f (x) = 2 x^{2} - 5 \sqrt{x}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 2 {(x + h)}^{2} - 5 \sqrt{x + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = 2 ((x + h) (x + h)) - 5 \sqrt{x + h}$

Step 2.1.2.1.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 2 (x (x + h) + h (x + h)) - 5 \sqrt{x + h}$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h (x + h)) - 5 \sqrt{x + h}$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h x + h \cdot h) - 5 \sqrt{x + h}$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 2 (x^{2} + x h + h x + h \cdot h) - 5 \sqrt{x + h}$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = 2 (x^{2} + x h + h x + h^{2}) - 5 \sqrt{x + h}$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = 2 (x^{2} + h x + h x + h^{2}) - 5 \sqrt{x + h}$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = 2 (x^{2} + 2 h x + h^{2}) - 5 \sqrt{x + h}$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = 2 x^{2} + 2 (2 h x) + 2 h^{2} - 5 \sqrt{x + h}$

Step 2.1.2.1.5

Multiply $2$ by $2$ .

$f (x + h) = 2 x^{2} + 4 h x + 2 h^{2} - 5 \sqrt{x + h}$

Step 2.1.2.2

The final answer is $2 x^{2} + 4 h x + 2 h^{2} - 5 \sqrt{x + h}$ .

$2 x^{2} + 4 h x + 2 h^{2} - 5 \sqrt{x + h}$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $x$ and $h$ .

$2 x^{2} + 4 h x + 2 h^{2} - 5 \sqrt{h + x}$

Step 2.2.2

Move $2 x^{2}$ .

$4 h x + 2 h^{2} + 2 x^{2} - 5 \sqrt{h + x}$

Step 2.2.3

Reorder $4 h x$ and $2 h^{2}$ .

$2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x}$

Step 2.3

Find the components of the definition.

$f (x + h) = 2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x}$

$f (x) = 2 x^{2} - 5 \sqrt{x}$

$f (x + h) = 2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x}$

$f (x) = 2 x^{2} - 5 \sqrt{x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x} - (2 x^{2} - 5 \sqrt{x})}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x} - (2 x^{2}) - (- 5 \sqrt{x})}{h}$

Step 4.2

Multiply $2$ by $- 1$ .

$\frac{2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x} - 2 x^{2} - (- 5 \sqrt{x})}{h}$

Step 4.3

Multiply $- 5$ by $- 1$ .

$\frac{2 h^{2} + 4 h x + 2 x^{2} - 5 \sqrt{h + x} - 2 x^{2} + 5 \sqrt{x}}{h}$

Step 4.4

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{2 h^{2} + 4 h x + 0 - 5 \sqrt{h + x} + 5 \sqrt{x}}{h}$

Step 4.5

Add $2 h^{2}$ and $0$ .

$\frac{2 h^{2} + 4 h x - 5 \sqrt{h + x} + 5 \sqrt{x}}{h}$

Step 5