Calculus Examples

$f (x) = 2 e^{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{x}]$ .

$2 \frac{d}{d x} [e^{x}]$

Step 1.2

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

$2 e^{x}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{x}]$ .

$f'' (x) = 2 \frac{d}{d x} (e^{x})$

Step 2.2

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

$f'' (x) = 2 e^{x}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$2 e^{x} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6