Calculus Examples

$1 - \arctan (e^{x})$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

$0 - \frac{d}{d x} [\arctan (e^{x})]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$0 - (\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [e^{x}])$

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

$0 - (\frac{1}{1 + {(e^{x})}^{2}} e^{x})$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$0 - (\frac{1}{1 + e^{x \cdot 2}} e^{x})$

Step 2.4.2

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

$0 - (\frac{1}{1 + e^{2 x}} e^{x})$

Step 2.5

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

$0 - \frac{e^{x}}{1 + e^{2 x}}$

Step 3

Subtract $\frac{e^{x}}{1 + e^{2 x}}$ from $0$ .

$- \frac{e^{x}}{1 + e^{2 x}}$