Calculus Examples

$x^{2} \arctan (e^{x})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \arctan (e^{x})$ .

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

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 $g (x) = e^{x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $e^{x}$ .

$x^{2} (\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [e^{x}]) + \arctan (e^{x}) \frac{d}{d x} [x^{2}]$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $e^{x}$ .

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

Step 3

Combine fractions.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Move $2$ to the left of $x$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

To write $\arctan (e^{x}) (2 x)$ as a fraction with a common denominator, multiply by $\frac{1 + e^{2 x}}{1 + e^{2 x}}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Simplify the numerator.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 8.2.1.2

Multiply $2$ by $1$ .

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

Step 8.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 8.2.2

Reorder factors in $x^{2} e^{x} + 2 \arctan (e^{x}) x + 2 \arctan (e^{x}) x e^{2 x}$ .

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

Step 8.3

Reorder terms.

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