Calculus Examples

$p = \frac{1000 - 10 x}{40 - x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (p) = \frac{d}{d x} (\frac{1000 - 10 x}{40 - x})$

Step 2

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

$p'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1000 - 10 x$ and $g (x) = 40 - x$ .

$\frac{(40 - x) \frac{d}{d x} [1000 - 10 x] - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $1000 - 10 x$ with respect to $x$ is $\frac{d}{d x} [1000] + \frac{d}{d x} [- 10 x]$ .

$\frac{(40 - x) (\frac{d}{d x} [1000] + \frac{d}{d x} [- 10 x]) - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.2

Since $1000$ is constant with respect to $x$ , the derivative of $1000$ with respect to $x$ is $0$ .

$\frac{(40 - x) (0 + \frac{d}{d x} [- 10 x]) - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- 10 x]$ .

$\frac{(40 - x) \frac{d}{d x} [- 10 x] - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.4

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$\frac{(40 - x) (- 10 \frac{d}{d x} [x]) - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.5

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

$\frac{(40 - x) (- 10 \cdot 1) - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Multiply $- 10$ by $1$ .

$\frac{(40 - x) \cdot - 10 - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.6.2

Move $- 10$ to the left of $40 - x$ .

$\frac{- 10 \cdot (40 - x) - (1000 - 10 x) \frac{d}{d x} [40 - x]}{{(40 - x)}^{2}}$

Step 3.2.7

By the Sum Rule, the derivative of $40 - x$ with respect to $x$ is $\frac{d}{d x} [40] + \frac{d}{d x} [- x]$ .

$\frac{- 10 (40 - x) - (1000 - 10 x) (\frac{d}{d x} [40] + \frac{d}{d x} [- x])}{{(40 - x)}^{2}}$

Step 3.2.8

Since $40$ is constant with respect to $x$ , the derivative of $40$ with respect to $x$ is $0$ .

$\frac{- 10 (40 - x) - (1000 - 10 x) (0 + \frac{d}{d x} [- x])}{{(40 - x)}^{2}}$

Step 3.2.9

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

$\frac{- 10 (40 - x) - (1000 - 10 x) \frac{d}{d x} [- x]}{{(40 - x)}^{2}}$

Step 3.2.10

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

$\frac{- 10 (40 - x) - (1000 - 10 x) (- \frac{d}{d x} [x])}{{(40 - x)}^{2}}$

Step 3.2.11

Multiply.

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Step 3.2.11.1

Multiply $- 1$ by $- 1$ .

$\frac{- 10 (40 - x) + 1 (1000 - 10 x) \frac{d}{d x} [x]}{{(40 - x)}^{2}}$

Step 3.2.11.2

Multiply $1000 - 10 x$ by $1$ .

$\frac{- 10 (40 - x) + (1000 - 10 x) \frac{d}{d x} [x]}{{(40 - x)}^{2}}$

Step 3.2.12

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

$\frac{- 10 (40 - x) + (1000 - 10 x) \cdot 1}{{(40 - x)}^{2}}$

Step 3.2.13

Multiply $1000 - 10 x$ by $1$ .

$\frac{- 10 (40 - x) + 1000 - 10 x}{{(40 - x)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{- 10 \cdot 40 - 10 (- x) + 1000 - 10 x}{{(40 - x)}^{2}}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $- 10$ by $40$ .

$\frac{- 400 - 10 (- x) + 1000 - 10 x}{{(40 - x)}^{2}}$

Step 3.3.2.1.2

Multiply $- 1$ by $- 10$ .

$\frac{- 400 + 10 x + 1000 - 10 x}{{(40 - x)}^{2}}$

Step 3.3.2.2

Combine the opposite terms in $- 400 + 10 x + 1000 - 10 x$ .

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Step 3.3.2.2.1

Subtract $10 x$ from $10 x$ .

$\frac{- 400 + 0 + 1000}{{(40 - x)}^{2}}$

Step 3.3.2.2.2

Add $- 400$ and $0$ .

$\frac{- 400 + 1000}{{(40 - x)}^{2}}$

Step 3.3.2.3

Add $- 400$ and $1000$ .

$\frac{600}{{(40 - x)}^{2}}$

Step 3.3.3

Reorder terms.

$\frac{600}{{(- x + 40)}^{2}}$

Step 4

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

$p' = \frac{600}{{(- x + 40)}^{2}}$

Step 5

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

$\frac{d p}{d x} = \frac{600}{{(- x + 40)}^{2}}$