Calculus Examples

$f (x) = x^{2} + 5 x^{3}$

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.

$f (x + h) = {(x + h)}^{2} + 5 {(x + h)}^{3}$

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

$f (x + h) = (x + h) (x + h) + 5 {(x + h)}^{3}$

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.

$f (x + h) = x (x + h) + h (x + h) + 5 {(x + h)}^{3}$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) + 5 {(x + h)}^{3}$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 5 {(x + h)}^{3}$

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 $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h + 5 {(x + h)}^{3}$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} + 5 {(x + h)}^{3}$

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 $h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} + 5 {(x + h)}^{3}$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 {(x + h)}^{3}$

Step 2.1.2.1.4

Use the Binomial Theorem.

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3})$

Step 2.1.2.1.5

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 x^{3} + 5 (3 x^{2} h) + 5 (3 x h^{2}) + 5 h^{3}$

Step 2.1.2.1.6

Simplify.

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Step 2.1.2.1.6.1

Multiply $3$ by $5$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 (x^{2} h) + 5 (3 x h^{2}) + 5 h^{3}$

Step 2.1.2.1.6.2

Multiply $3$ by $5$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 (x^{2} h) + 15 (x h^{2}) + 5 h^{3}$

Step 2.1.2.1.7

Remove parentheses.

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3}$

Step 2.1.2.2

The final answer is $x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3}$ .

$x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3}$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 h x^{2} + 15 x h^{2} + 5 h^{3}$

Step 2.2.2

Move $x$ .

$x^{2} + 2 h x + h^{2} + 5 x^{3} + 15 h x^{2} + 15 h^{2} x + 5 h^{3}$

Step 2.2.3

Move $x^{2}$ .

$2 h x + h^{2} + 5 x^{3} + 15 h x^{2} + 15 h^{2} x + 5 h^{3} + x^{2}$

Step 2.2.4

Move $2 h x$ .

$h^{2} + 5 x^{3} + 15 h x^{2} + 15 h^{2} x + 5 h^{3} + 2 h x + x^{2}$

Step 2.2.5

Move $h^{2}$ .

$5 x^{3} + 15 h x^{2} + 15 h^{2} x + 5 h^{3} + h^{2} + 2 h x + x^{2}$

Step 2.2.6

Move $5 x^{3}$ .

$15 h x^{2} + 15 h^{2} x + 5 h^{3} + 5 x^{3} + h^{2} + 2 h x + x^{2}$

Step 2.2.7

Move $15 h x^{2}$ .

$15 h^{2} x + 5 h^{3} + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2}$

Step 2.2.8

Reorder $15 h^{2} x$ and $5 h^{3}$ .

$5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2}$

Step 2.3

Find the components of the definition.

$f (x + h) = 5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2}$

$f (x) = x^{2} + 5 x^{3}$

$f (x + h) = 5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2}$

$f (x) = x^{2} + 5 x^{3}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2} - (x^{2} + 5 x^{3})}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2} - x^{2} - (5 x^{3})}{h}$

Step 4.1.2

Multiply $5$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} + h^{2} + 2 h x + x^{2} - x^{2} - 5 x^{3}}{h}$

Step 4.1.3

Subtract $5 x^{3}$ from $5 x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x + x^{2} - x^{2} + 0}{h}$

Step 4.1.4

Add $5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x + x^{2} - x^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x + x^{2} - x^{2}}{h}$

Step 4.1.5

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

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x + 0}{h}$

Step 4.1.6

Add $5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x}{h}$

Step 4.1.7

Factor $h$ out of $5 h^{3} + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x$ .

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Step 4.1.7.1

Factor $h$ out of $5 h^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + 15 h^{2} x + 15 h x^{2} + h^{2} + 2 h x}{h}$

Step 4.1.7.2

Factor $h$ out of $15 h^{2} x$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + 15 h x^{2} + h^{2} + 2 h x}{h}$

Step 4.1.7.3

Factor $h$ out of $15 h x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + h (15 x^{2}) + h^{2} + 2 h x}{h}$

Step 4.1.7.4

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

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + h (15 x^{2}) + h (h) + 2 h x}{h}$

Step 4.1.7.5

Factor $h$ out of $2 h x$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + h (15 x^{2}) + h (h) + h (2 x)}{h}$

Step 4.1.7.6

Factor $h$ out of $h (5 h^{2}) + h (15 h x)$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x) + h (15 x^{2}) + h (h) + h (2 x)}{h}$

Step 4.1.7.7

Factor $h$ out of $h (5 h^{2} + 15 h x) + h (15 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2}) + h (h) + h (2 x)}{h}$

Step 4.1.7.8

Factor $h$ out of $h (5 h^{2} + 15 h x + 15 x^{2}) + h (h)$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2} + h) + h (2 x)}{h}$

Step 4.1.7.9

Factor $h$ out of $h (5 h^{2} + 15 h x + 15 x^{2} + h) + h (2 x)$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2} + h + 2 x)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2} + h + 2 x)}{h}$

Step 4.2.1.2

Divide $5 h^{2} + 15 h x + 15 x^{2} + h + 2 x$ by $1$ .

$f' (x) = lim_{h \to 0} 5 h^{2} + 15 h x + 15 x^{2} + h + 2 x$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

$f' (x) = lim_{h \to 0} 5 h^{2} + 15 x h + 15 x^{2} + h + 2 x$

Step 4.2.2.2

Move $h$ .

$f' (x) = lim_{h \to 0} 5 h^{2} + 15 x h + 15 x^{2} + 2 x + h$

Step 4.2.2.3

Move $5 h^{2}$ .

$f' (x) = lim_{h \to 0} 15 x h + 15 x^{2} + 5 h^{2} + 2 x + h$

Step 4.2.2.4

Reorder $15 x h$ and $15 x^{2}$ .

$f' (x) = lim_{h \to 0} 15 x^{2} + 15 x h + 5 h^{2} + 2 x + h$

Step 5

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

$lim_{h \to 0} 15 x^{2} + lim_{h \to 0} 15 x h + lim_{h \to 0} 5 h^{2} + lim_{h \to 0} 2 x + lim_{h \to 0} h$

Step 6

Evaluate the limit of $15 x^{2}$ which is constant as $h$ approaches $0$ .

$15 x^{2} + lim_{h \to 0} 15 x h + lim_{h \to 0} 5 h^{2} + lim_{h \to 0} 2 x + lim_{h \to 0} h$

Step 7

Move the term $15 x$ outside of the limit because it is constant with respect to $h$ .

$15 x^{2} + 15 x lim_{h \to 0} h + lim_{h \to 0} 5 h^{2} + lim_{h \to 0} 2 x + lim_{h \to 0} h$

Step 8

Move the term $5$ outside of the limit because it is constant with respect to $h$ .

$15 x^{2} + 15 x lim_{h \to 0} h + 5 lim_{h \to 0} h^{2} + lim_{h \to 0} 2 x + lim_{h \to 0} h$

Step 9

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$15 x^{2} + 15 x lim_{h \to 0} h + 5 {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 2 x + lim_{h \to 0} h$

Step 10

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

$15 x^{2} + 15 x lim_{h \to 0} h + 5 {(lim_{h \to 0} h)}^{2} + 2 x + lim_{h \to 0} h$

Step 11

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 11.1

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

$15 x^{2} + 15 x \cdot 0 + 5 {(lim_{h \to 0} h)}^{2} + 2 x + lim_{h \to 0} h$

Step 11.2

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

$15 x^{2} + 15 x \cdot 0 + 5 \cdot 0^{2} + 2 x + lim_{h \to 0} h$

Step 11.3

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

$15 x^{2} + 15 x \cdot 0 + 5 \cdot 0^{2} + 2 x + 0$

Step 12

Simplify the answer.

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Step 12.1

Add $15 x^{2} + 15 x \cdot 0 + 5 \cdot 0^{2} + 2 x$ and $0$ .

$15 x^{2} + 15 x \cdot 0 + 5 \cdot 0^{2} + 2 x$

Step 12.2

Simplify each term.

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Step 12.2.1

Multiply $15 x \cdot 0$ .

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Step 12.2.1.1

Multiply $0$ by $15$ .

$15 x^{2} + 0 x + 5 \cdot 0^{2} + 2 x$

Step 12.2.1.2

Multiply $0$ by $x$ .

$15 x^{2} + 0 + 5 \cdot 0^{2} + 2 x$

Step 12.2.2

Raising $0$ to any positive power yields $0$ .

$15 x^{2} + 0 + 5 \cdot 0 + 2 x$

Step 12.2.3

Multiply $5$ by $0$ .

$15 x^{2} + 0 + 0 + 2 x$

Step 12.3

Combine the opposite terms in $15 x^{2} + 0 + 0 + 2 x$ .

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Step 12.3.1

Add $15 x^{2}$ and $0$ .

$15 x^{2} + 0 + 2 x$

Step 12.3.2

Add $15 x^{2}$ and $0$ .

$15 x^{2} + 2 x$

Step 13