Calculus Examples

$y = \frac{2100 - 5 x}{3}$

Step 1

Write $y = \frac{2100 - 5 x}{3}$ as a function.

$f (x) = \frac{2100 - 5 x}{3}$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

Factor $5$ out of $2100 - 5 x$ .

Tap for more steps...

Step 2.1.1

Factor $5$ out of $2100$ .

$\frac{d}{d x} [\frac{5 (420) - 5 x}{3}]$

Step 2.1.2

Factor $5$ out of $- 5 x$ .

$\frac{d}{d x} [\frac{5 (420) + 5 (- x)}{3}]$

Step 2.1.3

Factor $5$ out of $5 (420) + 5 (- x)$ .

$\frac{d}{d x} [\frac{5 (420 - x)}{3}]$

Step 2.2

Since $\frac{5}{3}$ is constant with respect to $x$ , the derivative of $\frac{5 (420 - x)}{3}$ with respect to $x$ is $\frac{5}{3} \frac{d}{d x} [420 - x]$ .

$\frac{5}{3} \frac{d}{d x} [420 - x]$

Step 2.3

By the Sum Rule, the derivative of $420 - x$ with respect to $x$ is $\frac{d}{d x} [420] + \frac{d}{d x} [- x]$ .

$\frac{5}{3} (\frac{d}{d x} [420] + \frac{d}{d x} [- x])$

Step 2.4

Since $420$ is constant with respect to $x$ , the derivative of $420$ with respect to $x$ is $0$ .

$\frac{5}{3} (0 + \frac{d}{d x} [- x])$

Step 2.5

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

$\frac{5}{3} \frac{d}{d x} [- x]$

Step 2.6

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

$\frac{5}{3} (- \frac{d}{d x} [x])$

Step 2.7

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

$\frac{5}{3} (- 1 \cdot 1)$

Step 2.8

Combine fractions.

Tap for more steps...

Step 2.8.1

Multiply $- 1$ by $1$ .

$\frac{5}{3} \cdot - 1$

Step 2.8.2

Combine $\frac{5}{3}$ and $- 1$ .

$\frac{5 \cdot - 1}{3}$

Step 2.8.3

Simplify the expression.

Tap for more steps...

Step 2.8.3.1

Multiply $5$ by $- 1$ .

$\frac{- 5}{3}$

Step 2.8.3.2

Move the negative in front of the fraction.

$- \frac{5}{3}$

Step 3

Since $- \frac{5}{3}$ is constant with respect to $x$ , the derivative of $- \frac{5}{3}$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 4

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

$- \frac{5}{3} = 0$

Step 5

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 6

No Local Extrema

Step 7