Calculus Examples

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

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 $g (x) = \arctan (x)$ .

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

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.2.1.2

Multiply $2$ by $1$ .

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

Step 6.2.1.3

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

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Step 6.2.1.3.1

Move $x^{2}$ .

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

Step 6.2.1.3.2

Multiply $x^{2}$ by $x$ .

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Step 6.2.1.3.2.1

Raise $x$ to the power of $1$ .

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

Step 6.2.1.3.2.2

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

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

Step 6.2.1.3.3

Add $2$ and $1$ .

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

Step 6.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 6.2.2

Reorder factors in $x^{2} + 2 \arctan (x) x + 2 \arctan (x) x^{3}$ .

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

Step 6.3

Reorder terms.

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