Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $g (x) = 4 x$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $4 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [4 x]$

Step 1.1.1.2

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

$e^{u} \frac{d}{d x} [4 x]$

Step 1.1.1.3

Replace all occurrences of $u$ with $4 x$ .

$e^{4 x} \frac{d}{d x} [4 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$e^{4 x} (4 \frac{d}{d x} [x])$

Step 1.1.2.2

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

$e^{4 x} (4 \cdot 1)$

Step 1.1.2.3

Simplify the expression.

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Step 1.1.2.3.1

Multiply $4$ by $1$ .

$e^{4 x} \cdot 4$

Step 1.1.2.3.2

Move $4$ to the left of $e^{4 x}$ .

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 e^{4 x}$ with respect to $x$ is $4 \frac{d}{d x} [e^{4 x}]$ .

$4 \frac{d}{d x} [e^{4 x}]$

Step 1.2.2

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 $g (x) = 4 x$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $4 x$ .

$4 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [4 x])$

Step 1.2.2.2

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

$4 (e^{u} \frac{d}{d x} [4 x])$

Step 1.2.2.3

Replace all occurrences of $u$ with $4 x$ .

$4 (e^{4 x} \frac{d}{d x} [4 x])$

Step 1.2.3

Differentiate.

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Step 1.2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 e^{4 x} (4 \frac{d}{d x} [x])$

Step 1.2.3.2

Multiply $4$ by $4$ .

$16 e^{4 x} \frac{d}{d x} [x]$

Step 1.2.3.3

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

$16 e^{4 x} \cdot 1$

Step 1.2.3.4

Multiply $16$ by $1$ .

$f'' (x) = 16 e^{4 x}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $16 e^{4 x}$ .

$16 e^{4 x}$

Step 2

Set the second derivative equal to $0$ then solve the equation $16 e^{4 x} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$16 e^{4 x} = 0$

Step 2.2

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

$\ln (16 e^{4 x}) = \ln (0)$

Step 2.3

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

Undefined

Step 2.4

There is no solution for $16 e^{4 x} = 0$

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points