Calculus Examples

$r (x) = 250 x \cdot e^{- 0.000425 \cdot 250}$

Step 1

Find the first derivative of the function.

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Step 1.1

Multiply $- 0.000425$ by $250$ .

$\frac{d}{d x} [250 x \cdot e^{- 0.10625}]$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [250 x \cdot \frac{1}{e^{0.10625}}]$

Step 1.3

Combine fractions.

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Step 1.3.1

Combine $250$ and $\frac{1}{e^{0.10625}}$ .

$\frac{d}{d x} [x \cdot \frac{250}{e^{0.10625}}]$

Step 1.3.2

Combine $x$ and $\frac{250}{e^{0.10625}}$ .

$\frac{d}{d x} [\frac{x \cdot 250}{e^{0.10625}}]$

Step 1.3.3

Move $250$ to the left of $x$ .

$\frac{d}{d x} [\frac{250 \cdot x}{e^{0.10625}}]$

Step 1.4

Since $\frac{250}{e^{0.10625}}$ is constant with respect to $x$ , the derivative of $\frac{250 x}{e^{0.10625}}$ with respect to $x$ is $\frac{250}{e^{0.10625}} \frac{d}{d x} [x]$ .

$\frac{250}{e^{0.10625}} \frac{d}{d x} [x]$

Step 1.5

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

$\frac{250}{e^{0.10625}} \cdot 1$

Step 1.6

Multiply $\frac{250}{e^{0.10625}}$ by $1$ .

$\frac{250}{e^{0.10625}}$

Step 2

Since $\frac{250}{e^{0.10625}}$ is constant with respect to $x$ , the derivative of $\frac{250}{e^{0.10625}}$ with respect to $x$ is $0$ .

$r'' (x) = 0$

Step 3

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

$\frac{250}{e^{0.10625}} = 0$

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