Calculus Examples

$y = 4 \arctan (3 x^{4})$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \arctan (3 x^{4})$ with respect to $x$ is $4 \frac{d}{d x} [\arctan (3 x^{4})]$ .

$4 \frac{d}{d x} [\arctan (3 x^{4})]$

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 $g (x) = 3 x^{4}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x^{4}$ .

$4 (\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [3 x^{4}])$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $3 x^{4}$ .

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

Step 3

Differentiate.

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Step 3.1

Factor $3$ out of $3 x^{4}$ .

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

Step 3.2

Combine fractions.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Apply the product rule to $3 (x^{4})$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

Multiply $4$ by $2$ .

$4 (\frac{1}{1 + 9 x^{8}} \frac{d}{d x} [3 x^{4}])$

Step 3.2.2

Combine $\frac{1}{1 + 9 x^{8}}$ and $4$ .

$\frac{4}{1 + 9 x^{8}} \frac{d}{d x} [3 x^{4}]$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$\frac{4}{1 + 9 x^{8}} (3 \frac{d}{d x} [x^{4}])$

Step 3.4

Combine fractions.

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Step 3.4.1

Combine $3$ and $\frac{4}{1 + 9 x^{8}}$ .

$\frac{3 \cdot 4}{1 + 9 x^{8}} \frac{d}{d x} [x^{4}]$

Step 3.4.2

Multiply $3$ by $4$ .

$\frac{12}{1 + 9 x^{8}} \frac{d}{d x} [x^{4}]$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{12}{1 + 9 x^{8}} (4 x^{3})$

Step 3.6

Combine fractions.

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Step 3.6.1

Combine $4$ and $\frac{12}{1 + 9 x^{8}}$ .

$\frac{4 \cdot 12}{1 + 9 x^{8}} x^{3}$

Step 3.6.2

Multiply $4$ by $12$ .

$\frac{48}{1 + 9 x^{8}} x^{3}$

Step 3.6.3

Combine $\frac{48}{1 + 9 x^{8}}$ and $x^{3}$ .

$\frac{48 x^{3}}{1 + 9 x^{8}}$

Step 3.6.4

Reorder terms.

$\frac{48 x^{3}}{9 x^{8} + 1}$