Calculus Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

Factor $3$ out of $3 x$ .

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $3 (x)$ .

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

Step 1.2.2

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

$\frac{d}{d x} [\arctan (9 x^{2})]$

Step 2

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 $g (x) = 9 x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $9 x^{2}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [9 x^{2}]$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $9 x^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

Factor $9$ out of $9 x^{2}$ .

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

Step 3.2

Simplify the expression.

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Step 3.2.1

Apply the product rule to $9 (x^{2})$ .

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

Step 3.2.2

Raise $9$ to the power of $2$ .

$\frac{1}{1 + 81 {(x^{2})}^{2}} \frac{d}{d x} [9 x^{2}]$

Step 3.2.3

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

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Step 3.2.3.1

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

$\frac{1}{1 + 81 x^{2 \cdot 2}} \frac{d}{d x} [9 x^{2}]$

Step 3.2.3.2

Multiply $2$ by $2$ .

$\frac{1}{1 + 81 x^{4}} \frac{d}{d x} [9 x^{2}]$

Step 3.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{1 + 81 x^{4}} (9 \frac{d}{d x} [x^{2}])$

Step 3.4

Combine $9$ and $\frac{1}{1 + 81 x^{4}}$ .

$\frac{9}{1 + 81 x^{4}} \frac{d}{d x} [x^{2}]$

Step 3.5

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

$\frac{9}{1 + 81 x^{4}} (2 x)$

Step 3.6

Combine fractions.

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Step 3.6.1

Combine $2$ and $\frac{9}{1 + 81 x^{4}}$ .

$\frac{2 \cdot 9}{1 + 81 x^{4}} x$

Step 3.6.2

Multiply $2$ by $9$ .

$\frac{18}{1 + 81 x^{4}} x$

Step 3.6.3

Combine $\frac{18}{1 + 81 x^{4}}$ and $x$ .

$\frac{18 x}{1 + 81 x^{4}}$

Step 3.6.4

Reorder terms.

$\frac{18 x}{81 x^{4} + 1}$