Calculus Examples

${(1 - u^{2})}^{5}$

Step 1

Use the Binomial Theorem.

$1^{5} + 5 \cdot 1^{4} (- u^{2}) + 10 \cdot 1^{3} {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2

Simplify each term.

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Step 2.1

One to any power is one.

$1 + 5 \cdot 1^{4} (- u^{2}) + 10 \cdot 1^{3} {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.2

One to any power is one.

$1 + 5 \cdot 1 (- u^{2}) + 10 \cdot 1^{3} {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.3

Multiply $5$ by $1$ .

$1 + 5 (- u^{2}) + 10 \cdot 1^{3} {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.4

Multiply $- 1$ by $5$ .

$1 - 5 u^{2} + 10 \cdot 1^{3} {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.5

One to any power is one.

$1 - 5 u^{2} + 10 \cdot 1 {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.6

Multiply $10$ by $1$ .

$1 - 5 u^{2} + 10 {(- u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.7

Apply the product rule to $- u^{2}$ .

$1 - 5 u^{2} + 10 ({(- 1)}^{2} {(u^{2})}^{2}) + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.8

Raise $- 1$ to the power of $2$ .

$1 - 5 u^{2} + 10 (1 {(u^{2})}^{2}) + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.9

Multiply ${(u^{2})}^{2}$ by $1$ .

$1 - 5 u^{2} + 10 {(u^{2})}^{2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.10

Multiply the exponents in ${(u^{2})}^{2}$ .

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Step 2.10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 - 5 u^{2} + 10 u^{2 \cdot 2} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.10.2

Multiply $2$ by $2$ .

$1 - 5 u^{2} + 10 u^{4} + 10 \cdot 1^{2} {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.11

One to any power is one.

$1 - 5 u^{2} + 10 u^{4} + 10 \cdot 1 {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.12

Multiply $10$ by $1$ .

$1 - 5 u^{2} + 10 u^{4} + 10 {(- u^{2})}^{3} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.13

Apply the product rule to $- u^{2}$ .

$1 - 5 u^{2} + 10 u^{4} + 10 ({(- 1)}^{3} {(u^{2})}^{3}) + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.14

Raise $- 1$ to the power of $3$ .

$1 - 5 u^{2} + 10 u^{4} + 10 (- {(u^{2})}^{3}) + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.15

Multiply the exponents in ${(u^{2})}^{3}$ .

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Step 2.15.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 - 5 u^{2} + 10 u^{4} + 10 (- u^{2 \cdot 3}) + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.15.2

Multiply $2$ by $3$ .

$1 - 5 u^{2} + 10 u^{4} + 10 (- u^{6}) + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.16

Multiply $- 1$ by $10$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 \cdot 1 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.17

Multiply $5$ by $1$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 {(- u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.18

Apply the product rule to $- u^{2}$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 ({(- 1)}^{4} {(u^{2})}^{4}) + {(- u^{2})}^{5}$

Step 2.19

Raise $- 1$ to the power of $4$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 (1 {(u^{2})}^{4}) + {(- u^{2})}^{5}$

Step 2.20

Multiply ${(u^{2})}^{4}$ by $1$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 {(u^{2})}^{4} + {(- u^{2})}^{5}$

Step 2.21

Multiply the exponents in ${(u^{2})}^{4}$ .

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Step 2.21.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{2 \cdot 4} + {(- u^{2})}^{5}$

Step 2.21.2

Multiply $2$ by $4$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} + {(- u^{2})}^{5}$

Step 2.22

Apply the product rule to $- u^{2}$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} + {(- 1)}^{5} {(u^{2})}^{5}$

Step 2.23

Raise $- 1$ to the power of $5$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} - {(u^{2})}^{5}$

Step 2.24

Multiply the exponents in ${(u^{2})}^{5}$ .

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Step 2.24.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} - u^{2 \cdot 5}$

Step 2.24.2

Multiply $2$ by $5$ .

$1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} - u^{10}$