Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\arctan (5 x^{2})$ .

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

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\arctan (5 x^{2})$ .

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

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 $g (x) = 5 x^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Factor $5$ out of $5 x^{2}$ .

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

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 $5 (x^{2})$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

Multiply $2$ by $2$ .

$\frac{1}{\arctan (5 x^{2})} (\frac{1}{1 + 25 x^{4}} \frac{d}{d x} [5 x^{2}])$

Step 3.2.2

Multiply $\frac{1}{1 + 25 x^{4}}$ by $\frac{1}{\arctan (5 x^{2})}$ .

$\frac{1}{(1 + 25 x^{4}) \arctan (5 x^{2})} \frac{d}{d x} [5 x^{2}]$

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{(1 + 25 x^{4}) \arctan (5 x^{2})} (5 \frac{d}{d x} [x^{2}])$

Step 3.4

Combine $5$ and $\frac{1}{(1 + 25 x^{4}) \arctan (5 x^{2})}$ .

$\frac{5}{(1 + 25 x^{4}) \arctan (5 x^{2})} \frac{d}{d x} [x^{2}]$

Step 3.5

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

$\frac{5}{(1 + 25 x^{4}) \arctan (5 x^{2})} (2 x)$

Step 3.6

Combine fractions.

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Step 3.6.1

Combine $2$ and $\frac{5}{(1 + 25 x^{4}) \arctan (5 x^{2})}$ .

$\frac{2 \cdot 5}{(1 + 25 x^{4}) \arctan (5 x^{2})} x$

Step 3.6.2

Multiply $2$ by $5$ .

$\frac{10}{(1 + 25 x^{4}) \arctan (5 x^{2})} x$

Step 3.6.3

Combine $\frac{10}{(1 + 25 x^{4}) \arctan (5 x^{2})}$ and $x$ .

$\frac{10 x}{(1 + 25 x^{4}) \arctan (5 x^{2})}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{10 x}{1 \arctan (5 x^{2}) + 25 x^{4} \arctan (5 x^{2})}$

Step 4.2

Multiply $\arctan (5 x^{2})$ by $1$ .

$\frac{10 x}{\arctan (5 x^{2}) + 25 x^{4} \arctan (5 x^{2})}$

Step 4.3

Reorder terms.

$\frac{10 x}{25 x^{4} \arctan (5 x^{2}) + \arctan (5 x^{2})}$

Step 4.4

Factor $\arctan (5 x^{2})$ out of $25 x^{4} \arctan (5 x^{2}) + \arctan (5 x^{2})$ .

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Step 4.4.1

Factor $\arctan (5 x^{2})$ out of $25 x^{4} \arctan (5 x^{2})$ .

$\frac{10 x}{\arctan (5 x^{2}) (25 x^{4}) + \arctan (5 x^{2})}$

Step 4.4.2

Multiply by $1$ .

$\frac{10 x}{\arctan (5 x^{2}) (25 x^{4}) + \arctan (5 x^{2}) \cdot 1}$

Step 4.4.3

Factor $\arctan (5 x^{2})$ out of $\arctan (5 x^{2}) (25 x^{4}) + \arctan (5 x^{2}) \cdot 1$ .

$\frac{10 x}{\arctan (5 x^{2}) (25 x^{4} + 1)}$