Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = \frac{e^{2 x}}{3}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{e^{2 x}}{3}$ .

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

Step 1.2

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

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [\frac{e^{2 x}}{3}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{e^{2 x}}{3}$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{e^{2 x}}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [e^{2 x}]$ .

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

Step 2.2

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

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

Step 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 $g (x) = 2 x$ .

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Step 3.1

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

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

Step 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 $a$ = $e$ .

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

Step 3.3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{e^{2 x}}{3}$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $3$ by $1$ .

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

Step 5.3.2

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

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Step 5.3.2.1

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

$\frac{2 e^{2 x}}{3 + 3 \frac{e^{2 x \cdot 2}}{3^{2}}}$

Step 5.3.2.2

Multiply $2$ by $2$ .

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

Step 5.3.3

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

$\frac{2 e^{2 x}}{3 + 3 \frac{e^{4 x}}{9}}$

Step 5.3.4

Combine $3$ and $\frac{e^{4 x}}{9}$ .

$\frac{2 e^{2 x}}{3 + \frac{3 e^{4 x}}{9}}$

Step 5.3.5

Cancel the common factor of $3$ and $9$ .

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Step 5.3.5.1

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

$\frac{2 e^{2 x}}{3 + \frac{3 (e^{4 x})}{9}}$

Step 5.3.5.2

Cancel the common factors.

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Step 5.3.5.2.1

Factor $3$ out of $9$ .

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

Step 5.3.5.2.2

Cancel the common factor.

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

Step 5.3.5.2.3

Rewrite the expression.

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

Step 5.4

Simplify the denominator.

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Step 5.4.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.4.2

Combine $3$ and $\frac{3}{3}$ .

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

Step 5.4.3

Combine the numerators over the common denominator.

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

Step 5.4.4

Multiply $3$ by $3$ .

$\frac{2 e^{2 x}}{\frac{9 + e^{4 x}}{3}}$

Step 5.5

Multiply the numerator by the reciprocal of the denominator.

$2 e^{2 x} \frac{3}{9 + e^{4 x}}$

Step 5.6

Multiply $2 e^{2 x} \frac{3}{9 + e^{4 x}}$ .

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Step 5.6.1

Combine $\frac{3}{9 + e^{4 x}}$ and $2$ .

$\frac{3 \cdot 2}{9 + e^{4 x}} e^{2 x}$

Step 5.6.2

Multiply $3$ by $2$ .

$\frac{6}{9 + e^{4 x}} e^{2 x}$

Step 5.6.3

Combine $\frac{6}{9 + e^{4 x}}$ and $e^{2 x}$ .

$\frac{6 e^{2 x}}{9 + e^{4 x}}$