Calculus Examples

$f (x) = 200 p r^{2} (1 - \frac{8 r}{3})$

Step 1

Since $200 r^{2} (1 - \frac{8 r}{3})$ is constant with respect to $p$ , the derivative of $200 p r^{2} (1 - \frac{8 r}{3})$ with respect to $p$ is $200 r^{2} (1 - \frac{8 r}{3}) \frac{d}{d p} [p]$ .

$200 r^{2} (1 - \frac{8 r}{3}) \frac{d}{d p} [p]$

Step 2

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

$200 r^{2} (1 - \frac{8 r}{3}) \cdot 1$

Step 3

Multiply $200$ by $1$ .

$200 r^{2} (1 - \frac{8 r}{3})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$200 r^{2} \cdot 1 + 200 r^{2} (- \frac{8 r}{3})$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $200$ by $1$ .

$200 r^{2} + 200 r^{2} (- \frac{8 r}{3})$

Step 4.2.2

Multiply $- 1$ by $200$ .

$200 r^{2} - 200 r^{2} \frac{8 r}{3}$

Step 4.2.3

Combine $- 200$ and $\frac{8 r}{3}$ .

$200 r^{2} + r^{2} \frac{- 200 (8 r)}{3}$

Step 4.2.4

Multiply $8$ by $- 200$ .

$200 r^{2} + r^{2} \frac{- 1600 r}{3}$

Step 4.2.5

Combine $r^{2}$ and $\frac{- 1600 r}{3}$ .

$200 r^{2} + \frac{r^{2} (- 1600 r)}{3}$

Step 4.2.6

Multiply $r^{2}$ by $r$ by adding the exponents.

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Step 4.2.6.1

Move $r$ .

$200 r^{2} + \frac{r \cdot r^{2} \cdot - 1600}{3}$

Step 4.2.6.2

Multiply $r$ by $r^{2}$ .

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Step 4.2.6.2.1

Raise $r$ to the power of $1$ .

$200 r^{2} + \frac{r^{1} r^{2} \cdot - 1600}{3}$

Step 4.2.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$200 r^{2} + \frac{r^{1 + 2} \cdot - 1600}{3}$

Step 4.2.6.3

Add $1$ and $2$ .

$200 r^{2} + \frac{r^{3} \cdot - 1600}{3}$

Step 4.2.7

Move $- 1600$ to the left of $r^{3}$ .

$200 r^{2} + \frac{- 1600 \cdot r^{3}}{3}$

Step 4.2.8

Move the negative in front of the fraction.

$200 r^{2} - \frac{1600 r^{3}}{3}$

Step 4.3

Reorder terms.

$- \frac{1600 r^{3}}{3} + 200 r^{2}$