Calculus Examples

$13 e^{- x}$

Step 1

Write $13 e^{- x}$ as a function.

$f (x) = 13 e^{- x}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Since $13$ is constant with respect to $x$ , the derivative of $13 e^{- x}$ with respect to $x$ is $13 \frac{d}{d x} [e^{- x}]$ .

$13 \frac{d}{d x} [e^{- x}]$

Step 2.1.1.2

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 $g (x) = - x$ .

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Step 2.1.1.2.1

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

$13 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x])$

Step 2.1.1.2.2

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

$13 (e^{u} \frac{d}{d x} [- x])$

Step 2.1.1.2.3

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

$13 (e^{- x} \frac{d}{d x} [- x])$

Step 2.1.1.3

Differentiate.

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Step 2.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$13 e^{- x} (- \frac{d}{d x} [x])$

Step 2.1.1.3.2

Multiply $- 1$ by $13$ .

$- 13 e^{- x} \frac{d}{d x} [x]$

Step 2.1.1.3.3

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

$- 13 e^{- x} \cdot 1$

Step 2.1.1.3.4

Multiply $- 13$ by $1$ .

$f' (x) = - 13 e^{- x}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13 e^{- x}$ with respect to $x$ is $- 13 \frac{d}{d x} [e^{- x}]$ .

$- 13 \frac{d}{d x} [e^{- x}]$

Step 2.1.2.2

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 $g (x) = - x$ .

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Step 2.1.2.2.1

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

$- 13 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x])$

Step 2.1.2.2.2

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

$- 13 (e^{u} \frac{d}{d x} [- x])$

Step 2.1.2.2.3

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

$- 13 (e^{- x} \frac{d}{d x} [- x])$

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- 13 e^{- x} (- \frac{d}{d x} [x])$

Step 2.1.2.3.2

Multiply $- 1$ by $- 13$ .

$13 e^{- x} \frac{d}{d x} [x]$

Step 2.1.2.3.3

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

$13 e^{- x} \cdot 1$

Step 2.1.2.3.4

Multiply $13$ by $1$ .

$f'' (x) = 13 e^{- x}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $13 e^{- x}$ .

$13 e^{- x}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $13 e^{- x} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$13 e^{- x} = 0$

Step 2.2.2

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

$\ln (13 e^{- x}) = \ln (0)$

Step 2.2.3

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

Undefined

Step 2.2.4

There is no solution for $13 e^{- x} = 0$

No solution

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

The graph is concave up because the second derivative is positive.

The graph is concave up

Step 5