Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arctan (e^{x^{4}}))$

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 $f (x) = \arctan (x)$ and $g (x) = e^{x^{4}}$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $e^{x^{4}}$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d x} [e^{x^{4}}]$

Step 3.1.2

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

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [e^{x^{4}}]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $e^{x^{4}}$ .

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

Step 3.2

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

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Step 3.2.1

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

$\frac{1}{1 + e^{x^{4} \cdot 2}} \frac{d}{d x} [e^{x^{4}}]$

Step 3.2.2

Move $2$ to the left of $x^{4}$ .

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{4}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{4}$ .

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

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.3

Replace all occurrences of $u_{2}$ with $x^{4}$ .

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

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Combine fractions.

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Step 3.4.3.1

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

$\frac{4 e^{x^{4}}}{1 + e^{2 x^{4}}} x^{3}$

Step 3.4.3.2

Combine $\frac{4 e^{x^{4}}}{1 + e^{2 x^{4}}}$ and $x^{3}$ .

$\frac{4 e^{x^{4}} x^{3}}{1 + e^{2 x^{4}}}$

Step 3.5

Simplify.

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Step 3.5.1

Reorder factors in $4 e^{x^{4}} x^{3}$ .

$\frac{4 x^{3} e^{x^{4}}}{1 + e^{2 x^{4}}}$

Step 3.5.2

Reorder terms.

$\frac{4 e^{x^{4}} x^{3}}{1 + e^{2 x^{4}}}$

Step 3.5.3

Reorder factors in $\frac{4 e^{x^{4}} x^{3}}{1 + e^{2 x^{4}}}$ .

$\frac{4 x^{3} e^{x^{4}}}{1 + e^{2 x^{4}}}$

Step 4

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

$y' = \frac{4 x^{3} e^{x^{4}}}{1 + e^{2 x^{4}}}$

Step 5

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

$\frac{d y}{d x} = \frac{4 x^{3} e^{x^{4}}}{1 + e^{2 x^{4}}}$