Calculus Examples

$f (x) = x (24 - 0.06 x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 24 - 0.06 x$ .

$x \frac{d}{d x} [24 - 0.06 x] + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $24 - 0.06 x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- 0.06 x]$ .

$x (\frac{d}{d x} [24] + \frac{d}{d x} [- 0.06 x]) + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.2

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- 0.06 x]) + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 0.06 x]$ .

$x \frac{d}{d x} [- 0.06 x] + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.4

Since $- 0.06$ is constant with respect to $x$ , the derivative of $- 0.06 x$ with respect to $x$ is $- 0.06 \frac{d}{d x} [x]$ .

$x (- 0.06 \frac{d}{d x} [x]) + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.5

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

$x (- 0.06 \cdot 1) + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.6

Simplify the expression.

Tap for more steps...

Step 2.6.1

Multiply $- 0.06$ by $1$ .

$x \cdot - 0.06 + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.6.2

Move $- 0.06$ to the left of $x$ .

$- 0.06 \cdot x + (24 - 0.06 x) \frac{d}{d x} [x]$

Step 2.7

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

$- 0.06 x + (24 - 0.06 x) \cdot 1$

Step 2.8

Simplify by adding terms.

Tap for more steps...

Step 2.8.1

Multiply $24 - 0.06 x$ by $1$ .

$- 0.06 x + 24 - 0.06 x$

Step 2.8.2

Subtract $0.06 x$ from $- 0.06 x$ .

$- 0.12 x + 24$