Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = u^{2}$ and $g (u) = 1 - u^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $1 - u^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d u} [1 - u^{2}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d u} [1 - u^{2}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $1 - u^{2}$ .

$2 (1 - u^{2}) \frac{d}{d u} [1 - u^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 - u^{2}$ with respect to $u$ is $\frac{d}{d u} [1] + \frac{d}{d u} [- u^{2}]$ .

$2 (1 - u^{2}) (\frac{d}{d u} [1] + \frac{d}{d u} [- u^{2}])$

Step 2.2

Since $1$ is constant with respect to $u$ , the derivative of $1$ with respect to $u$ is $0$ .

$2 (1 - u^{2}) (0 + \frac{d}{d u} [- u^{2}])$

Step 2.3

Add $0$ and $\frac{d}{d u} [- u^{2}]$ .

$2 (1 - u^{2}) \frac{d}{d u} [- u^{2}]$

Step 2.4

Since $- 1$ is constant with respect to $u$ , the derivative of $- u^{2}$ with respect to $u$ is $- \frac{d}{d u} [u^{2}]$ .

$2 (1 - u^{2}) (- \frac{d}{d u} [u^{2}])$

Step 2.5

Multiply $- 1$ by $2$ .

$- 2 (1 - u^{2}) \frac{d}{d u} [u^{2}]$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 2.7

Multiply $2$ by $- 2$ .

$- 4 (1 - u^{2}) u$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(- 4 \cdot 1 - 4 (- u^{2})) u$

Step 3.2

Apply the distributive property.

$- 4 \cdot 1 u - 4 (- u^{2}) u$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $- 4$ by $1$ .

$- 4 u - 4 (- u^{2}) u$

Step 3.3.2

Multiply $- 1$ by $- 4$ .

$- 4 u + 4 u^{2} u$

Step 3.3.3

Raise $u$ to the power of $1$ .

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

Step 3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 4 u + 4 u^{1 + 2}$

Step 3.3.5

Add $1$ and $2$ .

$- 4 u + 4 u^{3}$

Step 3.4

Reorder terms.

$4 u^{3} - 4 u$