Calculus Examples

$e^{x} - x^{3} \arctan (x)$

Step 1

By the Sum Rule, the derivative of $e^{x} - x^{3} \arctan (x)$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- x^{3} \arctan (x)]$ .

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- x^{3} \arctan (x)]$

Step 2

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

$e^{x} + \frac{d}{d x} [- x^{3} \arctan (x)]$

Step 3

Evaluate $\frac{d}{d x} [- x^{3} \arctan (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3} \arctan (x)$ with respect to $x$ is $- \frac{d}{d x} [x^{3} \arctan (x)]$ .

$e^{x} - \frac{d}{d x} [x^{3} \arctan (x)]$

Step 3.2

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

$e^{x} - (x^{3} \frac{d}{d x} [\arctan (x)] + \arctan (x) \frac{d}{d x} [x^{3}])$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

To write $\arctan (x) (3 x^{2})$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $\arctan (x)$ by $1$ .

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

Step 4.2.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 4.2.2.1

Move $x^{2}$ .

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

Step 4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.2.3

Add $2$ and $2$ .

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

Step 4.2.3

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.3

Reorder terms.

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