Calculus Examples

$p (x) = \frac{1}{5} \cdot (150 - x)$

Step 1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} (150 - x)$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [150 - x]$ .

$\frac{1}{5} \frac{d}{d x} [150 - x]$

Step 2

By the Sum Rule, the derivative of $150 - x$ with respect to $x$ is $\frac{d}{d x} [150] + \frac{d}{d x} [- x]$ .

$\frac{1}{5} (\frac{d}{d x} [150] + \frac{d}{d x} [- x])$

Step 3

Since $150$ is constant with respect to $x$ , the derivative of $150$ with respect to $x$ is $0$ .

$\frac{1}{5} (0 + \frac{d}{d x} [- x])$

Step 4

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{1}{5} \frac{d}{d x} [- x]$

Step 5

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

$\frac{1}{5} (- \frac{d}{d x} [x])$

Step 6

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

$\frac{1}{5} (- 1 \cdot 1)$

Step 7

Combine fractions.

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Step 7.1

Multiply $- 1$ by $1$ .

$\frac{1}{5} \cdot - 1$

Step 7.2

Combine $\frac{1}{5}$ and $- 1$ .

$\frac{- 1}{5}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{1}{5}$