Calculus Examples

${(x^{2} + a^{2})}^{n}$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} + a^{2}$ .

$\frac{d}{d u} [u^{n}] \frac{d}{d x} [x^{2} + a^{2}]$

Step 1.2

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

$n u^{n - 1} \frac{d}{d x} [x^{2} + a^{2}]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} + a^{2}$ .

$n {(x^{2} + a^{2})}^{n - 1} \frac{d}{d x} [x^{2} + a^{2}]$

Step 2

Differentiate.

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Step 2.1

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

$n {(x^{2} + a^{2})}^{n - 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [a^{2}])$

Step 2.2

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

$n {(x^{2} + a^{2})}^{n - 1} (2 x + \frac{d}{d x} [a^{2}])$

Step 2.3

Since $a^{2}$ is constant with respect to $x$ , the derivative of $a^{2}$ with respect to $x$ is $0$ .

$n {(x^{2} + a^{2})}^{n - 1} (2 x + 0)$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

$n {(x^{2} + a^{2})}^{n - 1} (2 x)$

Step 2.4.2

Reorder the factors of $n {(x^{2} + a^{2})}^{n - 1} (2 x)$ .

$2 n x {(x^{2} + a^{2})}^{n - 1}$