Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$e^{2 x} (2 \cdot 1)$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $2$ by $1$ .

$e^{2 x} \cdot 2$

Step 1.2.3.2

Move $2$ to the left of $e^{2 x}$ .

$2 e^{2 x}$

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 $2 e^{2 x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{2 x}]$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $2$ .

$f'' (x) = 4 e^{2 x} \frac{d}{d x} (x)$

Step 2.3.3

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

$f'' (x) = 4 e^{2 x} \cdot 1$

Step 2.3.4

Multiply $4$ by $1$ .

$f'' (x) = 4 e^{2 x}$

Step 3

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

$2 e^{2 x} = 0$

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