Calculus Examples

$\frac{4}{3} \cdot (p r^{3})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d r} [f (r) g (r)]$ is $f (r) \frac{d}{d r} [g (r)] + g (r) \frac{d}{d r} [f (r)]$ where $f (r) = \frac{4}{3}$ and $g (r) = p r^{3}$ .

$\frac{4}{3} \frac{d}{d r} [p r^{3}] + p r^{3} \frac{d}{d r} [\frac{4}{3}]$

Step 2

Differentiate.

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Step 2.1

Since $p$ is constant with respect to $r$ , the derivative of $p r^{3}$ with respect to $r$ is $p \frac{d}{d r} [r^{3}]$ .

$\frac{4}{3} (p \frac{d}{d r} [r^{3}]) + p r^{3} \frac{d}{d r} [\frac{4}{3}]$

Step 2.2

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

$\frac{4}{3} (p (3 r^{2})) + p r^{3} \frac{d}{d r} [\frac{4}{3}]$

Step 2.3

Reorder.

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Step 2.3.1

Move $3$ to the left of $p$ .

$\frac{4}{3} (3 \cdot p r^{2}) + p r^{3} \frac{d}{d r} [\frac{4}{3}]$

Step 2.3.2

Reorder the factors of $3 p r^{2}$ .

$\frac{4}{3} (3 r^{2} p) + p r^{3} \frac{d}{d r} [\frac{4}{3}]$

Step 2.4

Since $\frac{4}{3}$ is constant with respect to $r$ , the derivative of $\frac{4}{3}$ with respect to $r$ is $0$ .

$\frac{4}{3} (3 r^{2} p) + p r^{3} \cdot 0$

Step 3

Combine terms.

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Step 3.1

Combine $3$ and $\frac{4}{3}$ .

$\frac{3 \cdot 4}{3} (r^{2} p) + p r^{3} \cdot 0$

Step 3.2

Multiply $3$ by $4$ .

$\frac{12}{3} (r^{2} p) + p r^{3} \cdot 0$

Step 3.3

Combine $r^{2}$ and $\frac{12}{3}$ .

$\frac{r^{2} \cdot 12}{3} p + p r^{3} \cdot 0$

Step 3.4

Combine $\frac{r^{2} \cdot 12}{3}$ and $p$ .

$\frac{r^{2} \cdot 12 p}{3} + p r^{3} \cdot 0$

Step 3.5

Move $12$ to the left of $r^{2}$ .

$\frac{12 \cdot r^{2} p}{3} + p r^{3} \cdot 0$

Step 3.6

Cancel the common factor of $12$ and $3$ .

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Step 3.6.1

Factor $3$ out of $12 r^{2} p$ .

$\frac{3 (4 r^{2} p)}{3} + p r^{3} \cdot 0$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (4 r^{2} p)}{3 (1)} + p r^{3} \cdot 0$

Step 3.6.2.2

Cancel the common factor.

$\frac{3 (4 r^{2} p)}{3 \cdot 1} + p r^{3} \cdot 0$

Step 3.6.2.3

Rewrite the expression.

$\frac{4 r^{2} p}{1} + p r^{3} \cdot 0$

Step 3.6.2.4

Divide $4 r^{2} p$ by $1$ .

$4 r^{2} p + p r^{3} \cdot 0$

Step 3.7

Multiply $0$ by $p$ .

$4 r^{2} p + 0 r^{3}$

Step 3.8

Multiply $0$ by $r^{3}$ .

$4 r^{2} p + 0$

Step 3.9

Add $4 r^{2} p$ and $0$ .

$4 r^{2} p$