Calculus Examples

$f (x) = \frac{1}{2} e^{- x}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} \cdot e^{- x}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [e^{- x}]$ .

$\frac{1}{2} \frac{d}{d x} [e^{- x}]$

Step 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 1.1.2.1

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

$\frac{1}{2} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x])$

Step 1.1.2.2

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

$\frac{1}{2} (e^{u} \frac{d}{d x} [- x])$

Step 1.1.2.3

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

$\frac{1}{2} (e^{- x} \frac{d}{d x} [- x])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Combine $e^{- x}$ and $\frac{1}{2}$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Combine fractions.

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Step 1.1.3.4.1

Multiply $- 1$ by $1$ .

$\frac{e^{- x}}{2} \cdot - 1$

Step 1.1.3.4.2

Combine $\frac{e^{- x}}{2}$ and $- 1$ .

$\frac{e^{- x} \cdot - 1}{2}$

Step 1.1.3.4.3

Simplify the expression.

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Step 1.1.3.4.3.1

Move $- 1$ to the left of $e^{- x}$ .

$\frac{- 1 \cdot e^{- x}}{2}$

Step 1.1.3.4.3.2

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$\frac{- e^{- x}}{2}$

Step 1.1.3.4.3.3

Move the negative in front of the fraction.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{e^{- x}}{2}$ .

$- \frac{e^{- x}}{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{e^{- x}}{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{e^{- x}}{2} = 0$

Step 2.2

Set the numerator equal to zero.

$e^{- x} = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

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

Step 2.3.2

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

Undefined

Step 2.3.3

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

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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.

Step 4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found