Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{4}{3} \cdot (p {(r + h)}^{3}) - \frac{4}{3} \cdot (p r^{3})$ with respect to $r$ is $\frac{d}{d r} [\frac{4}{3} \cdot (p {(r + h)}^{3})] + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$ .

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Combine $\frac{p \cdot 4}{3}$ and ${(r + h)}^{3}$ .

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

Step 2.3

Move $4$ to the left of $p$ .

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

Step 2.4

Since $\frac{4 p}{3}$ is constant with respect to $r$ , the derivative of $\frac{4 p {(r + h)}^{3}}{3}$ with respect to $r$ is $\frac{4 p}{3} \frac{d}{d r} [{(r + h)}^{3}]$ .

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

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = r^{3}$ and $g (r) = r + h$ .

Tap for more steps...

Step 2.5.1

To apply the Chain Rule, set $u$ as $r + h$ .

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

Step 2.5.2

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

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

Step 2.5.3

Replace all occurrences of $u$ with $r + h$ .

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

Step 2.6

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

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

Step 2.7

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

$\frac{4 p}{3} (3 {(r + h)}^{2} (1 + \frac{d}{d r} [h])) + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.8

Since $h$ is constant with respect to $r$ , the derivative of $h$ with respect to $r$ is $0$ .

$\frac{4 p}{3} (3 {(r + h)}^{2} (1 + 0)) + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.9

Add $1$ and $0$ .

$\frac{4 p}{3} (3 {(r + h)}^{2} \cdot 1) + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.10

Multiply $3$ by $1$ .

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

Step 2.11

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

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

Step 2.12

Multiply $4$ by $3$ .

$\frac{12 p}{3} {(r + h)}^{2} + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.13

Combine $\frac{12 p}{3}$ and ${(r + h)}^{2}$ .

$\frac{12 p {(r + h)}^{2}}{3} + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.14

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

Tap for more steps...

Step 2.14.1

Factor $3$ out of $12 p {(r + h)}^{2}$ .

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

Step 2.14.2

Cancel the common factors.

Tap for more steps...

Step 2.14.2.1

Factor $3$ out of $3$ .

$\frac{3 (4 p {(r + h)}^{2})}{3 (1)} + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.14.2.2

Cancel the common factor.

$\frac{3 (4 p {(r + h)}^{2})}{3 \cdot 1} + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.14.2.3

Rewrite the expression.

$\frac{4 p {(r + h)}^{2}}{1} + \frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$

Step 2.14.2.4

Divide $4 p {(r + h)}^{2}$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d r} [- \frac{4}{3} \cdot (p r^{3})]$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Combine $r^{3}$ and $\frac{p \cdot 4}{3}$ .

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

Step 3.3

Move $4$ to the left of $p$ .

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

Step 3.4

Move $4$ to the left of $r^{3}$ .

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

Step 3.5

Since $- \frac{4 p}{3}$ is constant with respect to $r$ , the derivative of $- \frac{4 r^{3} p}{3}$ with respect to $r$ is $- \frac{4 p}{3} \frac{d}{d r} [r^{3}]$ .

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

Step 3.6

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

$4 p {(r + h)}^{2} - \frac{4 p}{3} (3 r^{2})$

Step 3.7

Multiply $3$ by $- 1$ .

$4 p {(r + h)}^{2} - 3 \frac{4 p}{3} r^{2}$

Step 3.8

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

$4 p {(r + h)}^{2} + \frac{- 3 (4 p)}{3} r^{2}$

Step 3.9

Multiply $4$ by $- 3$ .

$4 p {(r + h)}^{2} + \frac{- 12 p}{3} r^{2}$

Step 3.10

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

$4 p {(r + h)}^{2} + \frac{- 12 p r^{2}}{3}$

Step 3.11

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

Tap for more steps...

Step 3.11.1

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

$4 p {(r + h)}^{2} + \frac{3 (- 4 p r^{2})}{3}$

Step 3.11.2

Cancel the common factors.

Tap for more steps...

Step 3.11.2.1

Factor $3$ out of $3$ .

$4 p {(r + h)}^{2} + \frac{3 (- 4 p r^{2})}{3 (1)}$

Step 3.11.2.2

Cancel the common factor.

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

Step 3.11.2.3

Rewrite the expression.

$4 p {(r + h)}^{2} + \frac{- 4 p r^{2}}{1}$

Step 3.11.2.4

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

$4 p {(r + h)}^{2} - 4 p r^{2}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder terms.

$4 p {(r + h)}^{2} - 4 r^{2} p$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Rewrite ${(r + h)}^{2}$ as $(r + h) (r + h)$ .

$4 p ((r + h) (r + h)) - 4 r^{2} p$

Step 4.2.2

Expand $(r + h) (r + h)$ using the FOIL Method.

Tap for more steps...

Step 4.2.2.1

Apply the distributive property.

$4 p (r (r + h) + h (r + h)) - 4 r^{2} p$

Step 4.2.2.2

Apply the distributive property.

$4 p (r \cdot r + r h + h (r + h)) - 4 r^{2} p$

Step 4.2.2.3

Apply the distributive property.

$4 p (r \cdot r + r h + h r + h \cdot h) - 4 r^{2} p$

Step 4.2.3

Simplify and combine like terms.

Tap for more steps...

Step 4.2.3.1

Simplify each term.

Tap for more steps...

Step 4.2.3.1.1

Multiply $r$ by $r$ .

$4 p (r^{2} + r h + h r + h \cdot h) - 4 r^{2} p$

Step 4.2.3.1.2

Multiply $h$ by $h$ .

$4 p (r^{2} + r h + h r + h^{2}) - 4 r^{2} p$

Step 4.2.3.2

Add $r h$ and $h r$ .

Tap for more steps...

Step 4.2.3.2.1

Reorder $r$ and $h$ .

$4 p (r^{2} + h r + h r + h^{2}) - 4 r^{2} p$

Step 4.2.3.2.2

Add $h r$ and $h r$ .

$4 p (r^{2} + 2 h r + h^{2}) - 4 r^{2} p$

Step 4.2.4

Apply the distributive property.

$4 p r^{2} + 4 p (2 h r) + 4 p h^{2} - 4 r^{2} p$

Step 4.2.5

Multiply $2$ by $4$ .

$4 p r^{2} + 8 p h r + 4 p h^{2} - 4 r^{2} p$

Step 4.3

Combine the opposite terms in $4 p r^{2} + 8 p h r + 4 p h^{2} - 4 r^{2} p$ .

Tap for more steps...

Step 4.3.1

Reorder the factors in the terms $4 p r^{2}$ and $- 4 r^{2} p$ .

$4 r^{2} p + 8 p h r + 4 p h^{2} - 4 r^{2} p$

Step 4.3.2

Subtract $4 r^{2} p$ from $4 r^{2} p$ .

$8 p h r + 4 p h^{2} + 0$

Step 4.3.3

Add $8 p h r + 4 p h^{2}$ and $0$ .

$8 p h r + 4 p h^{2}$