Calculus Examples

$f (x) = 4 x^{2} - 5 x$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

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.

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

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 $(x + h) (x + h)$ .

$f (x + h) = 4 ((x + h) (x + h)) - 5 (x + h)$

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.

$f (x + h) = 4 (x (x + h) + h (x + h)) - 5 (x + h)$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 4 (x \cdot x + x h + h (x + h)) - 5 (x + h)$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 4 (x \cdot x + x h + h x + h \cdot h) - 5 (x + h)$

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 $x$ .

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

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

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

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 $h$ .

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

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.1.4

Apply the distributive property.

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

Step 2.1.2.1.5

Multiply $2$ by $4$ .

$f (x + h) = 4 x^{2} + 8 (h x) + 4 h^{2} - 5 (x + h)$

Step 2.1.2.1.6

Apply the distributive property.

$f (x + h) = 4 x^{2} + 8 h x + 4 h^{2} - 5 x - 5 h$

Step 2.1.2.2

The final answer is $4 x^{2} + 8 h x + 4 h^{2} - 5 x - 5 h$ .

$4 x^{2} + 8 h x + 4 h^{2} - 5 x - 5 h$

Step 2.2

Reorder.

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Step 2.2.1

Move $- 5 x$ .

$4 x^{2} + 8 h x + 4 h^{2} - 5 h - 5 x$

Step 2.2.2

Move $4 x^{2}$ .

$8 h x + 4 h^{2} + 4 x^{2} - 5 h - 5 x$

Step 2.2.3

Reorder $8 h x$ and $4 h^{2}$ .

$4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x$

$f (x) = 4 x^{2} - 5 x$

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x$

$f (x) = 4 x^{2} - 5 x$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x - (4 x^{2} - 5 x)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x - (4 x^{2}) - (- 5 x)}{h}$

Step 4.1.2

Multiply $4$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x - 4 x^{2} - (- 5 x)}{h}$

Step 4.1.3

Multiply $- 5$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x + 4 x^{2} - 5 h - 5 x - 4 x^{2} + 5 x}{h}$

Step 4.1.4

Subtract $4 x^{2}$ from $4 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x - 5 h - 5 x + 0 + 5 x}{h}$

Step 4.1.5

Add $4 h^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x - 5 h - 5 x + 5 x}{h}$

Step 4.1.6

Add $- 5 x$ and $5 x$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x - 5 h + 0}{h}$

Step 4.1.7

Add $4 h^{2} + 8 h x - 5 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{2} + 8 h x - 5 h}{h}$

Step 4.1.8

Factor $h$ out of $4 h^{2} + 8 h x - 5 h$ .

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Step 4.1.8.1

Factor $h$ out of $4 h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h) + 8 h x - 5 h}{h}$

Step 4.1.8.2

Factor $h$ out of $8 h x$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h) + h (8 x) - 5 h}{h}$

Step 4.1.8.3

Factor $h$ out of $- 5 h$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h) + h (8 x) + h \cdot - 5}{h}$

Step 4.1.8.4

Factor $h$ out of $h (4 h) + h (8 x)$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h + 8 x) + h \cdot - 5}{h}$

Step 4.1.8.5

Factor $h$ out of $h (4 h + 8 x) + h \cdot - 5$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h + 8 x - 5)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (4 h + 8 x - 5)}{h}$

Step 4.2.1.2

Divide $4 h + 8 x - 5$ by $1$ .

$f' (x) = lim_{h \to 0} 4 h + 8 x - 5$

Step 4.2.2

Reorder $4 h$ and $8 x$ .

$f' (x) = lim_{h \to 0} 8 x + 4 h - 5$

Step 5

Evaluate the limit.

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Step 5.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$lim_{h \to 0} 8 x + lim_{h \to 0} 4 h - lim_{h \to 0} 5$

Step 5.2

Evaluate the limit of $8 x$ which is constant as $h$ approaches $0$ .

$8 x + lim_{h \to 0} 4 h - lim_{h \to 0} 5$

Step 5.3

Move the term $4$ outside of the limit because it is constant with respect to $h$ .

$8 x + 4 lim_{h \to 0} h - lim_{h \to 0} 5$

Step 5.4

Evaluate the limit of $5$ which is constant as $h$ approaches $0$ .

$8 x + 4 lim_{h \to 0} h - 1 \cdot 5$

Step 6

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$8 x + 4 \cdot 0 - 1 \cdot 5$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $4$ by $0$ .

$8 x + 0 - 1 \cdot 5$

Step 7.1.2

Multiply $- 1$ by $5$ .

$8 x + 0 - 5$

Step 7.2

Add $8 x$ and $0$ .

$8 x - 5$

Step 8