Calculus Examples

$f (x) = x^{5}$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Use the Binomial Theorem.

$f (x + h) = x^{5} + 5 x^{4} h + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.1.2.2

The final answer is $x^{5} + 5 x^{4} h + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$ .

$x^{5} + 5 x^{4} h + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{4}$ .

$x^{5} + 5 h x^{4} + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.2.2

Move $x^{3}$ .

$x^{5} + 5 h x^{4} + 10 h^{2} x^{3} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.2.3

Move $x^{2}$ .

$x^{5} + 5 h x^{4} + 10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 x h^{4} + h^{5}$

Step 2.2.4

Move $x$ .

$x^{5} + 5 h x^{4} + 10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 h^{4} x + h^{5}$

Step 2.2.5

Move $x^{5}$ .

$5 h x^{4} + 10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 h^{4} x + h^{5} + x^{5}$

Step 2.2.6

Move $5 h x^{4}$ .

$10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 h^{4} x + h^{5} + 5 h x^{4} + x^{5}$

Step 2.2.7

Move $10 h^{2} x^{3}$ .

$10 h^{3} x^{2} + 5 h^{4} x + h^{5} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

Step 2.2.8

Move $10 h^{3} x^{2}$ .

$5 h^{4} x + h^{5} + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

Step 2.2.9

Reorder $5 h^{4} x$ and $h^{5}$ .

$h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

$f (x) = x^{5}$

$f (x + h) = h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

$f (x) = x^{5}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5} - (x^{5})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$f' (x) = lim_{h \to 0} \frac{h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + 0}{h}$

Step 4.1.2

Add $h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3

Factor $h$ out of $h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}$ .

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Step 4.1.3.1

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

$f' (x) = lim_{h \to 0} \frac{h \cdot h^{4} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3.2

Factor $h$ out of $5 h^{4} x$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4}) + h (5 h^{3} x) + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3.3

Factor $h$ out of $10 h^{3} x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4}) + h (5 h^{3} x) + h (10 h^{2} x^{2}) + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3.4

Factor $h$ out of $10 h^{2} x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4}) + h (5 h^{3} x) + h (10 h^{2} x^{2}) + h (10 h x^{3}) + 5 h x^{4}}{h}$

Step 4.1.3.5

Factor $h$ out of $5 h x^{4}$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4}) + h (5 h^{3} x) + h (10 h^{2} x^{2}) + h (10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.6

Factor $h$ out of $h (h^{4}) + h (5 h^{3} x)$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4} + 5 h^{3} x) + h (10 h^{2} x^{2}) + h (10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.7

Factor $h$ out of $h (h^{4} + 5 h^{3} x) + h (10 h^{2} x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2}) + h (10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.8

Factor $h$ out of $h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2}) + h (10 h x^{3})$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.9

Factor $h$ out of $h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3}) + h (5 x^{4})$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4})}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4})}{h}$

Step 4.2.1.2

Divide $h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}$ by $1$ .

$f' (x) = lim_{h \to 0} h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h^{3}$ .

$f' (x) = lim_{h \to 0} h^{4} + 5 x h^{3} + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}$

Step 4.2.2.2

Move $h^{2}$ .

$f' (x) = lim_{h \to 0} h^{4} + 5 x h^{3} + 10 x^{2} h^{2} + 10 h x^{3} + 5 x^{4}$

Step 4.2.2.3

Move $h$ .

$f' (x) = lim_{h \to 0} h^{4} + 5 x h^{3} + 10 x^{2} h^{2} + 10 x^{3} h + 5 x^{4}$

Step 4.2.2.4

Move $h^{4}$ .

$f' (x) = lim_{h \to 0} 5 x h^{3} + 10 x^{2} h^{2} + 10 x^{3} h + 5 x^{4} + h^{4}$

Step 4.2.2.5

Move $5 x h^{3}$ .

$f' (x) = lim_{h \to 0} 10 x^{2} h^{2} + 10 x^{3} h + 5 x^{4} + 5 x h^{3} + h^{4}$

Step 4.2.2.6

Move $10 x^{2} h^{2}$ .

$f' (x) = lim_{h \to 0} 10 x^{3} h + 5 x^{4} + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}$

Step 4.2.2.7

Reorder $10 x^{3} h$ and $5 x^{4}$ .

$f' (x) = lim_{h \to 0} 5 x^{4} + 10 x^{3} h + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}$

Step 5

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

$lim_{h \to 0} 5 x^{4} + lim_{h \to 0} 10 x^{3} h + lim_{h \to 0} 10 x^{2} h^{2} + lim_{h \to 0} 5 x h^{3} + lim_{h \to 0} h^{4}$

Step 6

Evaluate the limit of $5 x^{4}$ which is constant as $h$ approaches $0$ .

$5 x^{4} + lim_{h \to 0} 10 x^{3} h + lim_{h \to 0} 10 x^{2} h^{2} + lim_{h \to 0} 5 x h^{3} + lim_{h \to 0} h^{4}$

Step 7

Move the term $10 x^{3}$ outside of the limit because it is constant with respect to $h$ .

$5 x^{4} + 10 x^{3} lim_{h \to 0} h + lim_{h \to 0} 10 x^{2} h^{2} + lim_{h \to 0} 5 x h^{3} + lim_{h \to 0} h^{4}$

Step 8

Move the term $10 x^{2}$ outside of the limit because it is constant with respect to $h$ .

$5 x^{4} + 10 x^{3} lim_{h \to 0} h + 10 x^{2} lim_{h \to 0} h^{2} + lim_{h \to 0} 5 x h^{3} + lim_{h \to 0} h^{4}$

Step 9

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

$5 x^{4} + 10 x^{3} lim_{h \to 0} h + 10 x^{2} {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 5 x h^{3} + lim_{h \to 0} h^{4}$

Step 10

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

$5 x^{4} + 10 x^{3} lim_{h \to 0} h + 10 x^{2} {(lim_{h \to 0} h)}^{2} + 5 x lim_{h \to 0} h^{3} + lim_{h \to 0} h^{4}$

Step 11

Move the exponent $3$ from $h^{3}$ outside the limit using the Limits Power Rule.

$5 x^{4} + 10 x^{3} lim_{h \to 0} h + 10 x^{2} {(lim_{h \to 0} h)}^{2} + 5 x {(lim_{h \to 0} h)}^{3} + lim_{h \to 0} h^{4}$

Step 12

Move the exponent $4$ from $h^{4}$ outside the limit using the Limits Power Rule.

$5 x^{4} + 10 x^{3} lim_{h \to 0} h + 10 x^{2} {(lim_{h \to 0} h)}^{2} + 5 x {(lim_{h \to 0} h)}^{3} + {(lim_{h \to 0} h)}^{4}$

Step 13

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

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Step 13.1

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

$5 x^{4} + 10 x^{3} \cdot 0 + 10 x^{2} {(lim_{h \to 0} h)}^{2} + 5 x {(lim_{h \to 0} h)}^{3} + {(lim_{h \to 0} h)}^{4}$

Step 13.2

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

$5 x^{4} + 10 x^{3} \cdot 0 + 10 x^{2} \cdot 0^{2} + 5 x {(lim_{h \to 0} h)}^{3} + {(lim_{h \to 0} h)}^{4}$

Step 13.3

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

$5 x^{4} + 10 x^{3} \cdot 0 + 10 x^{2} \cdot 0^{2} + 5 x \cdot 0^{3} + {(lim_{h \to 0} h)}^{4}$

Step 13.4

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

$5 x^{4} + 10 x^{3} \cdot 0 + 10 x^{2} \cdot 0^{2} + 5 x \cdot 0^{3} + 0^{4}$

Step 14

Simplify the answer.

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Step 14.1

Simplify each term.

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Step 14.1.1

Multiply $10 x^{3} \cdot 0$ .

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Step 14.1.1.1

Multiply $0$ by $10$ .

$5 x^{4} + 0 x^{3} + 10 x^{2} \cdot 0^{2} + 5 x \cdot 0^{3} + 0^{4}$

Step 14.1.1.2

Multiply $0$ by $x^{3}$ .

$5 x^{4} + 0 + 10 x^{2} \cdot 0^{2} + 5 x \cdot 0^{3} + 0^{4}$

Step 14.1.2

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

$5 x^{4} + 0 + 10 x^{2} \cdot 0 + 5 x \cdot 0^{3} + 0^{4}$

Step 14.1.3

Multiply $10 x^{2} \cdot 0$ .

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Step 14.1.3.1

Multiply $0$ by $10$ .

$5 x^{4} + 0 + 0 x^{2} + 5 x \cdot 0^{3} + 0^{4}$

Step 14.1.3.2

Multiply $0$ by $x^{2}$ .

$5 x^{4} + 0 + 0 + 5 x \cdot 0^{3} + 0^{4}$

Step 14.1.4

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

$5 x^{4} + 0 + 0 + 5 x \cdot 0 + 0^{4}$

Step 14.1.5

Multiply $5 x \cdot 0$ .

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Step 14.1.5.1

Multiply $0$ by $5$ .

$5 x^{4} + 0 + 0 + 0 x + 0^{4}$

Step 14.1.5.2

Multiply $0$ by $x$ .

$5 x^{4} + 0 + 0 + 0 + 0^{4}$

Step 14.1.6

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

$5 x^{4} + 0 + 0 + 0 + 0$

Step 14.2

Combine the opposite terms in $5 x^{4} + 0 + 0 + 0 + 0$ .

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Step 14.2.1

Add $5 x^{4}$ and $0$ .

$5 x^{4} + 0 + 0 + 0$

Step 14.2.2

Add $5 x^{4}$ and $0$ .

$5 x^{4} + 0 + 0$

Step 14.2.3

Add $5 x^{4}$ and $0$ .

$5 x^{4} + 0$

Step 14.2.4

Add $5 x^{4}$ and $0$ .

$5 x^{4}$

Step 15