Calculus Examples

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

Step 1

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$2^{x} \ln (2)$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$f'' (x) = \ln (2) (2^{x} \ln (2))$

Step 2.3

Raise $\ln (2)$ to the power of $1$ .

$f'' (x) = 2^{x} ((\ln (2)) \ln (2))$

Step 2.4

Raise $\ln (2)$ to the power of $1$ .

$f'' (x) = 2^{x} ((\ln (2)) (\ln (2)))$

Step 2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = 2^{x} {(\ln (2))}^{1 + 1}$

Step 2.6

Add $1$ and $1$ .

$f'' (x) = 2^{x} \ln^{2} (2)$

Step 3

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

$2^{x} \ln (2) = 0$

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