Calculus Examples

$16 x + 16 e^{x}$

Step 1

Write $16 x + 16 e^{x}$ as a function.

$f (x) = 16 x + 16 e^{x}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $16 x + 16 e^{x}$ with respect to $x$ is $\frac{d}{d x} [16 x] + \frac{d}{d x} [16 e^{x}]$ .

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

Step 2.1.2

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 2.1.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

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

Step 2.1.2.2

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

$16 \cdot 1 + \frac{d}{d x} [16 e^{x}]$

Step 2.1.2.3

Multiply $16$ by $1$ .

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

Step 2.1.3

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

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Step 2.1.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 e^{x}$ with respect to $x$ is $16 \frac{d}{d x} [e^{x}]$ .

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

Step 2.1.3.2

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

$16 + 16 e^{x}$

Step 2.1.4

Reorder terms.

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

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $16 e^{x} + 16$ with respect to $x$ is $\frac{d}{d x} [16 e^{x}] + \frac{d}{d x} [16]$ .

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

Step 2.2.2

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

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Step 2.2.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 e^{x}$ with respect to $x$ is $16 \frac{d}{d x} [e^{x}]$ .

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

Step 2.2.2.2

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

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

Step 2.2.3

Differentiate using the Constant Rule.

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Step 2.2.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$16 e^{x} + 0$

Step 2.2.3.2

Add $16 e^{x}$ and $0$ .

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

Step 2.3

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

$16 e^{x}$

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

$16 e^{x} = 0$

Step 3.2

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

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

Step 3.3

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

Undefined

Step 3.4

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

No solution

Step 4

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

No Inflection Points