Calculus Examples

$p = \frac{1}{12} x^{2} - 20 x + 1200$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (p) = \frac{d}{d x} (\frac{1}{12} x^{2} - 20 x + 1200)$

Step 2

The derivative of $p$ with respect to $x$ is $p'$ .

$p'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $\frac{1}{12} x^{2} - 20 x + 1200$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{12} x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$ .

$\frac{d}{d x} [\frac{1}{12} x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{1}{12} x^{2}]$ .

Tap for more steps...

Step 3.2.1

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{1}{12} x^{2}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{12} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{12} (2 x) + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.3

Combine $2$ and $\frac{1}{12}$ .

$\frac{2}{12} x + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.4

Combine $\frac{2}{12}$ and $x$ .

$\frac{2 x}{12} + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.5

Cancel the common factor of $2$ and $12$ .

Tap for more steps...

Step 3.2.5.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{12} + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.5.2

Cancel the common factors.

Tap for more steps...

Step 3.2.5.2.1

Factor $2$ out of $12$ .

$\frac{2 x}{2 \cdot 6} + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.5.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 6} + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.2.5.2.3

Rewrite the expression.

$\frac{x}{6} + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [1200]$

Step 3.3

Evaluate $\frac{d}{d x} [- 20 x]$ .

Tap for more steps...

Step 3.3.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x$ with respect to $x$ is $- 20 \frac{d}{d x} [x]$ .

$\frac{x}{6} - 20 \frac{d}{d x} [x] + \frac{d}{d x} [1200]$

Step 3.3.2

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

$\frac{x}{6} - 20 \cdot 1 + \frac{d}{d x} [1200]$

Step 3.3.3

Multiply $- 20$ by $1$ .

$\frac{x}{6} - 20 + \frac{d}{d x} [1200]$

Step 3.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.4.1

Since $1200$ is constant with respect to $x$ , the derivative of $1200$ with respect to $x$ is $0$ .

$\frac{x}{6} - 20 + 0$

Step 3.4.2

Add $\frac{x}{6} - 20$ and $0$ .

$\frac{x}{6} - 20$

Step 4

Reform the equation by setting the left side equal to the right side.

$p' = \frac{x}{6} - 20$

Step 5

Replace $p'$ with $\frac{d p}{d x}$ .

$\frac{d p}{d x} = \frac{x}{6} - 20$