Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\arctan (2 x^{3})$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\arctan (2 x^{3})]$

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\arctan (2 x^{3})$ .

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

Step 3.2

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 $g (x) = 2 x^{3}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

Combine fractions.

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Step 3.3.2.1

Simplify the expression.

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

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

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

Step 3.3.2.1.3

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 3.3.2.1.3.1

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

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

Step 3.3.2.1.3.2

Multiply $3$ by $2$ .

$\frac{1}{\arctan (2 x^{3})} (\frac{1}{1 + 4 x^{6}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2.2

Multiply $\frac{1}{1 + 4 x^{6}}$ by $\frac{1}{\arctan (2 x^{3})}$ .

$\frac{1}{(1 + 4 x^{6}) \arctan (2 x^{3})} \frac{d}{d x} [2 x^{3}]$

Step 3.3.3

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

$\frac{1}{(1 + 4 x^{6}) \arctan (2 x^{3})} (2 \frac{d}{d x} [x^{3}])$

Step 3.3.4

Combine $2$ and $\frac{1}{(1 + 4 x^{6}) \arctan (2 x^{3})}$ .

$\frac{2}{(1 + 4 x^{6}) \arctan (2 x^{3})} \frac{d}{d x} [x^{3}]$

Step 3.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{2}{(1 + 4 x^{6}) \arctan (2 x^{3})} (3 x^{2})$

Step 3.3.6

Combine fractions.

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Step 3.3.6.1

Combine $3$ and $\frac{2}{(1 + 4 x^{6}) \arctan (2 x^{3})}$ .

$\frac{3 \cdot 2}{(1 + 4 x^{6}) \arctan (2 x^{3})} x^{2}$

Step 3.3.6.2

Multiply $3$ by $2$ .

$\frac{6}{(1 + 4 x^{6}) \arctan (2 x^{3})} x^{2}$

Step 3.3.6.3

Combine $\frac{6}{(1 + 4 x^{6}) \arctan (2 x^{3})}$ and $x^{2}$ .

$\frac{6 x^{2}}{(1 + 4 x^{6}) \arctan (2 x^{3})}$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$\frac{6 x^{2}}{1 \arctan (2 x^{3}) + 4 x^{6} \arctan (2 x^{3})}$

Step 3.4.2

Multiply $\arctan (2 x^{3})$ by $1$ .

$\frac{6 x^{2}}{\arctan (2 x^{3}) + 4 x^{6} \arctan (2 x^{3})}$

Step 3.4.3

Reorder terms.

$\frac{6 x^{2}}{4 x^{6} \arctan (2 x^{3}) + \arctan (2 x^{3})}$

Step 3.4.4

Factor $\arctan (2 x^{3})$ out of $4 x^{6} \arctan (2 x^{3}) + \arctan (2 x^{3})$ .

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Step 3.4.4.1

Factor $\arctan (2 x^{3})$ out of $4 x^{6} \arctan (2 x^{3})$ .

$\frac{6 x^{2}}{\arctan (2 x^{3}) (4 x^{6}) + \arctan (2 x^{3})}$

Step 3.4.4.2

Multiply by $1$ .

$\frac{6 x^{2}}{\arctan (2 x^{3}) (4 x^{6}) + \arctan (2 x^{3}) \cdot 1}$

Step 3.4.4.3

Factor $\arctan (2 x^{3})$ out of $\arctan (2 x^{3}) (4 x^{6}) + \arctan (2 x^{3}) \cdot 1$ .

$\frac{6 x^{2}}{\arctan (2 x^{3}) (4 x^{6} + 1)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{6 x^{2}}{\arctan (2 x^{3}) (4 x^{6} + 1)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{6 x^{2}}{\arctan (2 x^{3}) (4 x^{6} + 1)}$