Calculus Examples

$f (x) = \arctan (3 x^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = 3 x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x^{2}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [3 x^{2}]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d x} [3 x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $3 x^{2}$ .

$\frac{1}{1 + {(3 x^{2})}^{2}} \frac{d}{d x} [3 x^{2}]$

Step 2

Differentiate.

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Step 2.1

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

$\frac{1}{1 + {(3 (x^{2}))}^{2}} \frac{d}{d x} [3 x^{2}]$

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $3 (x^{2})$ .

$\frac{1}{1 + 3^{2} {(x^{2})}^{2}} \frac{d}{d x} [3 x^{2}]$

Step 2.2.2

Raise $3$ to the power of $2$ .

$\frac{1}{1 + 9 {(x^{2})}^{2}} \frac{d}{d x} [3 x^{2}]$

Step 2.2.3

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

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Step 2.2.3.1

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

$\frac{1}{1 + 9 x^{2 \cdot 2}} \frac{d}{d x} [3 x^{2}]$

Step 2.2.3.2

Multiply $2$ by $2$ .

$\frac{1}{1 + 9 x^{4}} \frac{d}{d x} [3 x^{2}]$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{1 + 9 x^{4}} (3 \frac{d}{d x} [x^{2}])$

Step 2.4

Combine $3$ and $\frac{1}{1 + 9 x^{4}}$ .

$\frac{3}{1 + 9 x^{4}} \frac{d}{d x} [x^{2}]$

Step 2.5

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

$\frac{3}{1 + 9 x^{4}} (2 x)$

Step 2.6

Combine fractions.

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Step 2.6.1

Combine $2$ and $\frac{3}{1 + 9 x^{4}}$ .

$\frac{2 \cdot 3}{1 + 9 x^{4}} x$

Step 2.6.2

Multiply $2$ by $3$ .

$\frac{6}{1 + 9 x^{4}} x$

Step 2.6.3

Combine $\frac{6}{1 + 9 x^{4}}$ and $x$ .

$\frac{6 x}{1 + 9 x^{4}}$

Step 2.6.4

Reorder terms.

$\frac{6 x}{9 x^{4} + 1}$