Calculus Examples

$r = 70 (30 x - x^{\frac{3}{2}})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (r) = \frac{d}{d x} (70 (30 x - x^{\frac{3}{2}}))$

Step 2

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

$r'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $70$ is constant with respect to $x$ , the derivative of $70 (30 x - x^{\frac{3}{2}})$ with respect to $x$ is $70 \frac{d}{d x} [30 x - x^{\frac{3}{2}}]$ .

$70 \frac{d}{d x} [30 x - x^{\frac{3}{2}}]$

Step 3.2

By the Sum Rule, the derivative of $30 x - x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [30 x] + \frac{d}{d x} [- x^{\frac{3}{2}}]$ .

$70 (\frac{d}{d x} [30 x] + \frac{d}{d x} [- x^{\frac{3}{2}}])$

Step 3.3

Since $30$ is constant with respect to $x$ , the derivative of $30 x$ with respect to $x$ is $30 \frac{d}{d x} [x]$ .

$70 (30 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{\frac{3}{2}}])$

Step 3.4

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

$70 (30 \cdot 1 + \frac{d}{d x} [- x^{\frac{3}{2}}])$

Step 3.5

Multiply $30$ by $1$ .

$70 (30 + \frac{d}{d x} [- x^{\frac{3}{2}}])$

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{3}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$70 (30 - \frac{d}{d x} [x^{\frac{3}{2}}])$

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$70 (30 - (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 3.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$70 (30 - (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.9

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

$70 (30 - (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.10

Combine the numerators over the common denominator.

$70 (30 - (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}))$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $2$ .

$70 (30 - (\frac{3}{2} x^{\frac{3 - 2}{2}}))$

Step 3.11.2

Subtract $2$ from $3$ .

$70 (30 - (\frac{3}{2} x^{\frac{1}{2}}))$

Step 3.12

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$70 (30 - \frac{3 x^{\frac{1}{2}}}{2})$

Step 3.13

Simplify.

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Step 3.13.1

Apply the distributive property.

$70 \cdot 30 + 70 (- \frac{3 x^{\frac{1}{2}}}{2})$

Step 3.13.2

Combine terms.

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Step 3.13.2.1

Multiply $70$ by $30$ .

$2100 + 70 (- \frac{3 x^{\frac{1}{2}}}{2})$

Step 3.13.2.2

Multiply $- 1$ by $70$ .

$2100 - 70 \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.13.2.3

Combine $- 70$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$2100 + \frac{- 70 (3 x^{\frac{1}{2}})}{2}$

Step 3.13.2.4

Multiply $3$ by $- 70$ .

$2100 + \frac{- 210 x^{\frac{1}{2}}}{2}$

Step 3.13.2.5

Factor $2$ out of $- 210 x^{\frac{1}{2}}$ .

$2100 + \frac{2 (- 105 x^{\frac{1}{2}})}{2}$

Step 3.13.2.6

Cancel the common factors.

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Step 3.13.2.6.1

Factor $2$ out of $2$ .

$2100 + \frac{2 (- 105 x^{\frac{1}{2}})}{2 (1)}$

Step 3.13.2.6.2

Cancel the common factor.

$2100 + \frac{2 (- 105 x^{\frac{1}{2}})}{2 \cdot 1}$

Step 3.13.2.6.3

Rewrite the expression.

$2100 + \frac{- 105 x^{\frac{1}{2}}}{1}$

Step 3.13.2.6.4

Divide $- 105 x^{\frac{1}{2}}$ by $1$ .

$2100 - 105 x^{\frac{1}{2}}$

Step 3.13.3

Reorder terms.

$- 105 x^{\frac{1}{2}} + 2100$

Step 4

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

$r' = - 105 x^{\frac{1}{2}} + 2100$

Step 5

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

$\frac{d r}{d x} = - 105 x^{\frac{1}{2}} + 2100$