Calculus Examples

$e^{x}$

Step 1

Write $e^{x}$ as a function.

$f (x) = e^{x}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

$e^{x}$

Step 2.2

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

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Step 2.2.1

Set the second derivative equal to $0$ .

$e^{x} = 0$

Step 2.2.2

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

$\ln (e^{x}) = \ln (0)$

Step 2.2.3

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.2.4

There is no solution for $e^{x} = 0$

No solution

Step 3

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

The graph is concave up because the second derivative is positive.

The graph is concave up

Step 5