Calculus Examples

$y = {(\arctan (5 x))}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \arctan (5 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\arctan (5 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\arctan (5 x)]$

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 $n = 2$ .

$2 u_{1} \frac{d}{d x} [\arctan (5 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\arctan (5 x)$ .

$2 \arctan (5 x) \frac{d}{d x} [\arctan (5 x)]$

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 $g (x) = 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $5 x$ .

$2 \arctan (5 x) (\frac{d}{d u_{2}} [\arctan (u_{2})] \frac{d}{d x} [5 x])$

Step 2.2

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

$2 \arctan (5 x) (\frac{1}{1 + {u_{2}}^{2}} \frac{d}{d x} [5 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $5 x$ .

$2 \arctan (5 x) (\frac{1}{1 + {(5 x)}^{2}} \frac{d}{d x} [5 x])$

Step 3

Differentiate.

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Step 3.1

Factor $5$ out of $5 x$ .

$2 \arctan (5 x) (\frac{1}{1 + {(5 (x))}^{2}} \frac{d}{d x} [5 x])$

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 $5 (x)$ .

$2 \arctan (5 x) (\frac{1}{1 + 5^{2} x^{2}} \frac{d}{d x} [5 x])$

Step 3.2.1.2

Raise $5$ to the power of $2$ .

$2 \arctan (5 x) (\frac{1}{1 + 25 x^{2}} \frac{d}{d x} [5 x])$

Step 3.2.2

Combine $\frac{1}{1 + 25 x^{2}}$ and $2$ .

$\frac{2}{1 + 25 x^{2}} \arctan (5 x) \frac{d}{d x} [5 x]$

Step 3.2.3

Combine $\frac{2}{1 + 25 x^{2}}$ and $\arctan (5 x)$ .

$\frac{2 \arctan (5 x)}{1 + 25 x^{2}} \frac{d}{d x} [5 x]$

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{2 \arctan (5 x)}{1 + 25 x^{2}} (5 \frac{d}{d x} [x])$

Step 3.4

Combine fractions.

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Step 3.4.1

Combine $5$ and $\frac{2 \arctan (5 x)}{1 + 25 x^{2}}$ .

$\frac{5 (2 \arctan (5 x))}{1 + 25 x^{2}} \frac{d}{d x} [x]$

Step 3.4.2

Multiply $2$ by $5$ .

$\frac{10 \arctan (5 x)}{1 + 25 x^{2}} \frac{d}{d x} [x]$

Step 3.5

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

$\frac{10 \arctan (5 x)}{1 + 25 x^{2}} \cdot 1$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $\frac{10 \arctan (5 x)}{1 + 25 x^{2}}$ by $1$ .

$\frac{10 \arctan (5 x)}{1 + 25 x^{2}}$

Step 3.6.2

Reorder terms.

$\frac{10 \arctan (5 x)}{25 x^{2} + 1}$