Calculus Examples

$y = (1 + x^{2}) \arctan (x) - x$

Step 1

By the Sum Rule, the derivative of $(1 + x^{2}) \arctan (x) - x$ with respect to $x$ is $\frac{d}{d x} [(1 + x^{2}) \arctan (x)] + \frac{d}{d x} [- x]$ .

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

Step 2

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

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Step 2.1

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 $f (x) = 1 + x^{2}$ and $g (x) = \arctan (x)$ .

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

Step 2.2

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

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

Step 2.3

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

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

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 + x^{2}) \frac{1}{1 + x^{2}} + \arctan (x) (0 + \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- x]$

Step 2.5

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

$(1 + x^{2}) \frac{1}{1 + x^{2}} + \arctan (x) (0 + 2 x) + \frac{d}{d x} [- x]$

Step 2.6

Add $0$ and $2 x$ .

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

Step 3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Cancel the common factor of $1 + x^{2}$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$1 + 2 x \arctan (x) - 1$

Step 4.3

Combine the opposite terms in $1 + 2 x \arctan (x) - 1$ .

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Step 4.3.1

Subtract $1$ from $1$ .

$2 x \arctan (x) + 0$

Step 4.3.2

Add $2 x \arctan (x)$ and $0$ .

$2 x \arctan (x)$