Calculus Examples

$f (t) = 2 + e^{t}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $2 + e^{t}$ with respect to $t$ is $\frac{d}{d t} [2] + \frac{d}{d t} [e^{t}]$ .

$\frac{d}{d t} [2] + \frac{d}{d t} [e^{t}]$

Step 1.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [e^{t}]$

Step 1.2

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

$0 + e^{t}$

Step 1.3

Add $0$ and $e^{t}$ .

$e^{t}$

Step 2

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

$f'' (t) = e^{t}$

Step 3

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

$e^{t} = 0$

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