Calculus Examples

$e^{x} - x$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

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

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- x]$

Step 1.1.2

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

$e^{x} + \frac{d}{d x} [- x]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$e^{x} - \frac{d}{d x} [x]$

Step 1.1.3.2

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

$e^{x} - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$f' (x) = e^{x} - 1$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $e^{x} - 1$ .

$e^{x} - 1$

Step 2

Set the first derivative equal to $0$ then solve the equation $e^{x} - 1 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$e^{x} - 1 = 0$

Step 2.2

Add $1$ to both sides of the equation.

$e^{x} = 1$

Step 2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (1)$

Step 2.4

Expand the left side.

Tap for more steps...

Step 2.4.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (1)$

Step 2.4.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (1)$

Step 2.4.3

Multiply $x$ by $1$ .

$x = \ln (1)$

Step 2.5

The natural logarithm of $1$ is $0$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $e^{x} - x$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

$e^{0} - (0)$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Anything raised to $0$ is $1$ .

$1 - (0)$

Step 4.1.2.2

Subtract $0$ from $1$ .

$1$

Step 4.2

List all of the points.

$(0, 1)$

Step 5