Calculus Examples

$h (t) = {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{3}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = {(t + 1)}^{\frac{2}{3}}$ and $g (t) = {(2 t^{2} - 1)}^{3}$ .

${(t + 1)}^{\frac{2}{3}} \frac{d}{d t} [{(2 t^{2} - 1)}^{3}] + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{3}$ and $g (t) = 2 t^{2} - 1$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $2 t^{2} - 1$ .

${(t + 1)}^{\frac{2}{3}} (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d t} [2 t^{2} - 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 2.2

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

${(t + 1)}^{\frac{2}{3}} (3 {u_{1}}^{2} \frac{d}{d t} [2 t^{2} - 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $2 t^{2} - 1$ .

${(t + 1)}^{\frac{2}{3}} (3 {(2 t^{2} - 1)}^{2} \frac{d}{d t} [2 t^{2} - 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Move $3$ to the left of ${(t + 1)}^{\frac{2}{3}}$ .

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

Step 3.2

By the Sum Rule, the derivative of $2 t^{2} - 1$ with respect to $t$ is $\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [- 1]$ .

$3 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} (\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [- 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3.3

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

$3 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} (2 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3.4

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

$3 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} (2 (2 t) + \frac{d}{d t} [- 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3.5

Multiply $2$ by $2$ .

$3 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} (4 t + \frac{d}{d t} [- 1]) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3.6

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$3 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} (4 t + 0) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3.7

Simplify the expression.

Tap for more steps...

Step 3.7.1

Add $4 t$ and $0$ .

$3 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} (4 t) + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 3.7.2

Multiply $4$ by $3$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} \frac{d}{d t} [{(t + 1)}^{\frac{2}{3}}]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{2}{3}}$ and $g (t) = t + 1$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $t + 1$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{2}{3}}] \frac{d}{d t} [t + 1])$

Step 4.2

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

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {u_{2}}^{\frac{2}{3} - 1} \frac{d}{d t} [t + 1])$

Step 4.3

Replace all occurrences of $u_{2}$ with $t + 1$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{\frac{2}{3} - 1} \frac{d}{d t} [t + 1])$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d t} [t + 1])$

Step 6

Combine $- 1$ and $\frac{3}{3}$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d t} [t + 1])$

Step 7

Combine the numerators over the common denominator.

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d t} [t + 1])$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $3$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{\frac{2 - 3}{3}} \frac{d}{d t} [t + 1])$

Step 8.2

Subtract $3$ from $2$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{\frac{- 1}{3}} \frac{d}{d t} [t + 1])$

Step 9

Combine fractions.

Tap for more steps...

Step 9.1

Move the negative in front of the fraction.

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3} {(t + 1)}^{- \frac{1}{3}} \frac{d}{d t} [t + 1])$

Step 9.2

Combine $\frac{2}{3}$ and ${(t + 1)}^{- \frac{1}{3}}$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2 {(t + 1)}^{- \frac{1}{3}}}{3} \frac{d}{d t} [t + 1])$

Step 9.3

Move ${(t + 1)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + {(2 t^{2} - 1)}^{3} (\frac{2}{3 {(t + 1)}^{\frac{1}{3}}} \frac{d}{d t} [t + 1])$

Step 9.4

Combine $\frac{2}{3 {(t + 1)}^{\frac{1}{3}}}$ and ${(2 t^{2} - 1)}^{3}$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}} \frac{d}{d t} [t + 1]$

Step 10

By the Sum Rule, the derivative of $t + 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [1]$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}} (\frac{d}{d t} [t] + \frac{d}{d t} [1])$

Step 11

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

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}} (1 + \frac{d}{d t} [1])$

Step 12

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}} (1 + 0)$

Step 13

Simplify the expression.

Tap for more steps...

Step 13.1

Add $1$ and $0$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}} \cdot 1$

Step 13.2

Multiply $\frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$ by $1$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 14

To write $12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t$ as a fraction with a common denominator, multiply by $\frac{3 {(t + 1)}^{\frac{1}{3}}}{3 {(t + 1)}^{\frac{1}{3}}}$ .

$12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t \cdot \frac{3 {(t + 1)}^{\frac{1}{3}}}{3 {(t + 1)}^{\frac{1}{3}}} + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 15

Combine $12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t$ and $\frac{3 {(t + 1)}^{\frac{1}{3}}}{3 {(t + 1)}^{\frac{1}{3}}}$ .

$\frac{12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t (3 {(t + 1)}^{\frac{1}{3}})}{3 {(t + 1)}^{\frac{1}{3}}} + \frac{2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 16

Combine the numerators over the common denominator.

$\frac{12 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t (3 {(t + 1)}^{\frac{1}{3}}) + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 17

Multiply $3$ by $12$ .

$\frac{36 {(t + 1)}^{\frac{2}{3}} {(2 t^{2} - 1)}^{2} t {(t + 1)}^{\frac{1}{3}} + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 18

Multiply ${(t + 1)}^{\frac{2}{3}}$ by ${(t + 1)}^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 18.1

Move ${(t + 1)}^{\frac{1}{3}}$ .

$\frac{36 ({(t + 1)}^{\frac{1}{3}} {(t + 1)}^{\frac{2}{3}}) {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 18.2

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

$\frac{36 {(t + 1)}^{\frac{1}{3} + \frac{2}{3}} {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 18.3

Combine the numerators over the common denominator.

$\frac{36 {(t + 1)}^{\frac{1 + 2}{3}} {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 18.4

Add $1$ and $2$ .

$\frac{36 {(t + 1)}^{\frac{3}{3}} {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 18.5

Divide $3$ by $3$ .

$\frac{36 {(t + 1)}^{1} {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 19

Simplify $36 {(t + 1)}^{1} {(2 t^{2} - 1)}^{2} t$ .

$\frac{36 (t + 1) {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20

Simplify.

Tap for more steps...

Step 20.1

Apply the distributive property.

$\frac{(36 t + 36 \cdot 1) {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2

Simplify the numerator.

Tap for more steps...

Step 20.2.1

Factor ${(2 t^{2} - 1)}^{2}$ out of $(36 t + 36 \cdot 1) {(2 t^{2} - 1)}^{2} t + 2 {(2 t^{2} - 1)}^{3}$ .

Tap for more steps...

Step 20.2.1.1

Factor ${(2 t^{2} - 1)}^{2}$ out of $(36 t + 36 \cdot 1) {(2 t^{2} - 1)}^{2} t$ .

$\frac{{(2 t^{2} - 1)}^{2} ((36 t + 36 \cdot 1) t) + 2 {(2 t^{2} - 1)}^{3}}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.1.2

Factor ${(2 t^{2} - 1)}^{2}$ out of $2 {(2 t^{2} - 1)}^{3}$ .

$\frac{{(2 t^{2} - 1)}^{2} ((36 t + 36 \cdot 1) t) + {(2 t^{2} - 1)}^{2} (2 (2 t^{2} - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.1.3

Factor ${(2 t^{2} - 1)}^{2}$ out of ${(2 t^{2} - 1)}^{2} ((36 t + 36 \cdot 1) t) + {(2 t^{2} - 1)}^{2} (2 (2 t^{2} - 1))$ .

$\frac{{(2 t^{2} - 1)}^{2} ((36 t + 36 \cdot 1) t + 2 (2 t^{2} - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.2

Multiply $36$ by $1$ .

$\frac{{(2 t^{2} - 1)}^{2} ((36 t + 36) t + 2 (2 t^{2} - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.3

Apply the distributive property.

$\frac{{(2 t^{2} - 1)}^{2} (36 t \cdot t + 36 t + 2 (2 t^{2} - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.4

Multiply $t$ by $t$ by adding the exponents.

Tap for more steps...

Step 20.2.4.1

Move $t$ .

$\frac{{(2 t^{2} - 1)}^{2} (36 (t \cdot t) + 36 t + 2 (2 t^{2} - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.4.2

Multiply $t$ by $t$ .

$\frac{{(2 t^{2} - 1)}^{2} (36 t^{2} + 36 t + 2 (2 t^{2} - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.5

Apply the distributive property.

$\frac{{(2 t^{2} - 1)}^{2} (36 t^{2} + 36 t + 2 (2 t^{2}) + 2 \cdot - 1)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.6

Multiply $2$ by $2$ .

$\frac{{(2 t^{2} - 1)}^{2} (36 t^{2} + 36 t + 4 t^{2} + 2 \cdot - 1)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.7

Multiply $2$ by $- 1$ .

$\frac{{(2 t^{2} - 1)}^{2} (36 t^{2} + 36 t + 4 t^{2} - 2)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.8

Add $36 t^{2}$ and $4 t^{2}$ .

$\frac{{(2 t^{2} - 1)}^{2} (40 t^{2} + 36 t - 2)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.9

Factor $2$ out of $40 t^{2} + 36 t - 2$ .

Tap for more steps...

Step 20.2.9.1

Factor $2$ out of $40 t^{2}$ .

$\frac{{(2 t^{2} - 1)}^{2} (2 (20 t^{2}) + 36 t - 2)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.9.2

Factor $2$ out of $36 t$ .

$\frac{{(2 t^{2} - 1)}^{2} (2 (20 t^{2}) + 2 (18 t) - 2)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.9.3

Factor $2$ out of $- 2$ .

$\frac{{(2 t^{2} - 1)}^{2} (2 (20 t^{2}) + 2 (18 t) + 2 (- 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.9.4

Factor $2$ out of $2 (20 t^{2}) + 2 (18 t)$ .

$\frac{{(2 t^{2} - 1)}^{2} (2 (20 t^{2} + 18 t) + 2 (- 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.2.9.5

Factor $2$ out of $2 (20 t^{2} + 18 t) + 2 (- 1)$ .

$\frac{{(2 t^{2} - 1)}^{2} (2 (20 t^{2} + 18 t - 1))}{3 {(t + 1)}^{\frac{1}{3}}}$

$\frac{{(2 t^{2} - 1)}^{2} \cdot 2 (20 t^{2} + 18 t - 1)}{3 {(t + 1)}^{\frac{1}{3}}}$

Step 20.3

Move $2$ to the left of ${(2 t^{2} - 1)}^{2}$ .

$\frac{2 {(2 t^{2} - 1)}^{2} (20 t^{2} + 18 t - 1)}{3 {(t + 1)}^{\frac{1}{3}}}$