Calculus Examples

$\frac{5}{t^{\frac{1}{5}}} - \frac{8}{t^{\frac{3}{2}}}$

Step 1

By the Sum Rule, the derivative of $\frac{5}{t^{\frac{1}{5}}} - \frac{8}{t^{\frac{3}{2}}}$ with respect to $t$ is $\frac{d}{d t} [\frac{5}{t^{\frac{1}{5}}}] + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$ .

$\frac{d}{d t} [\frac{5}{t^{\frac{1}{5}}}] + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2

Evaluate $\frac{d}{d t} [\frac{5}{t^{\frac{1}{5}}}]$ .

Tap for more steps...

Step 2.1

Since $5$ is constant with respect to $t$ , the derivative of $\frac{5}{t^{\frac{1}{5}}}$ with respect to $t$ is $5 \frac{d}{d t} [\frac{1}{t^{\frac{1}{5}}}]$ .

$5 \frac{d}{d t} [\frac{1}{t^{\frac{1}{5}}}] + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.2

Rewrite $\frac{1}{t^{\frac{1}{5}}}$ as ${(t^{\frac{1}{5}})}^{- 1}$ .

$5 \frac{d}{d t} [{(t^{\frac{1}{5}})}^{- 1}] + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{\frac{1}{5}}$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $t^{\frac{1}{5}}$ .

$5 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d t} [t^{\frac{1}{5}}]) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.3.2

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

$5 (- {u_{1}}^{- 2} \frac{d}{d t} [t^{\frac{1}{5}}]) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $t^{\frac{1}{5}}$ .

$5 (- {(t^{\frac{1}{5}})}^{- 2} \frac{d}{d t} [t^{\frac{1}{5}}]) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{5}$ .

$5 (- {(t^{\frac{1}{5}})}^{- 2} (\frac{1}{5} t^{\frac{1}{5} - 1})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.5

Multiply the exponents in ${(t^{\frac{1}{5}})}^{- 2}$ .

Tap for more steps...

Step 2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 (- t^{\frac{1}{5} \cdot - 2} (\frac{1}{5} t^{\frac{1}{5} - 1})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.5.2

Combine $\frac{1}{5}$ and $- 2$ .

$5 (- t^{\frac{- 2}{5}} (\frac{1}{5} t^{\frac{1}{5} - 1})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.5.3

Move the negative in front of the fraction.

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{\frac{1}{5} - 1})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{\frac{1}{5} - 1 \cdot \frac{5}{5}})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.7

Combine $- 1$ and $\frac{5}{5}$ .

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.8

Combine the numerators over the common denominator.

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{\frac{1 - 1 \cdot 5}{5}})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.9

Simplify the numerator.

Tap for more steps...

Step 2.9.1

Multiply $- 1$ by $5$ .

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{\frac{1 - 5}{5}})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.9.2

Subtract $5$ from $1$ .

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{\frac{- 4}{5}})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.10

Move the negative in front of the fraction.

$5 (- t^{- \frac{2}{5}} (\frac{1}{5} t^{- \frac{4}{5}})) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.11

Combine $\frac{1}{5}$ and $t^{- \frac{4}{5}}$ .

$5 (- t^{- \frac{2}{5}} \frac{t^{- \frac{4}{5}}}{5}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.12

Combine $\frac{t^{- \frac{4}{5}}}{5}$ and $t^{- \frac{2}{5}}$ .

$5 (- \frac{t^{- \frac{4}{5}} t^{- \frac{2}{5}}}{5}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.13

Multiply $t^{- \frac{4}{5}}$ by $t^{- \frac{2}{5}}$ by adding the exponents.

Tap for more steps...

Step 2.13.1

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

$5 (- \frac{t^{- \frac{4}{5} - \frac{2}{5}}}{5}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.13.2

Combine the numerators over the common denominator.

$5 (- \frac{t^{\frac{- 4 - 2}{5}}}{5}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.13.3

Subtract $2$ from $- 4$ .

$5 (- \frac{t^{\frac{- 6}{5}}}{5}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.13.4

Move the negative in front of the fraction.

$5 (- \frac{t^{- \frac{6}{5}}}{5}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.14

Move $t^{- \frac{6}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (- \frac{1}{5 t^{\frac{6}{5}}}) + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.15

Multiply $- 1$ by $5$ .

$- 5 \frac{1}{5 t^{\frac{6}{5}}} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.16

Combine $- 5$ and $\frac{1}{5 t^{\frac{6}{5}}}$ .

$\frac{- 5}{5 t^{\frac{6}{5}}} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.17

Factor $5$ out of $- 5$ .

$\frac{5 \cdot - 1}{5 t^{\frac{6}{5}}} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.18

Cancel the common factors.

Tap for more steps...

Step 2.18.1

Factor $5$ out of $5 t^{\frac{6}{5}}$ .

$\frac{5 \cdot - 1}{5 (t^{\frac{6}{5}})} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.18.2

Cancel the common factor.

$\frac{5 \cdot - 1}{5 t^{\frac{6}{5}}} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.18.3

Rewrite the expression.

$\frac{- 1}{t^{\frac{6}{5}}} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 2.19

Move the negative in front of the fraction.

$- \frac{1}{t^{\frac{6}{5}}} + \frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$

Step 3

Evaluate $\frac{d}{d t} [- \frac{8}{t^{\frac{3}{2}}}]$ .

Tap for more steps...

Step 3.1

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 \frac{d}{d t} [\frac{1}{t^{\frac{3}{2}}}]$

Step 3.2

Rewrite $\frac{1}{t^{\frac{3}{2}}}$ as ${(t^{\frac{3}{2}})}^{- 1}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 \frac{d}{d t} [{(t^{\frac{3}{2}})}^{- 1}]$

Step 3.3

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

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $t^{\frac{3}{2}}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d t} [t^{\frac{3}{2}}])$

Step 3.3.2

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- {u_{2}}^{- 2} \frac{d}{d t} [t^{\frac{3}{2}}])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $t^{\frac{3}{2}}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- {(t^{\frac{3}{2}})}^{- 2} \frac{d}{d t} [t^{\frac{3}{2}}])$

Step 3.4

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- {(t^{\frac{3}{2}})}^{- 2} (\frac{3}{2} t^{\frac{3}{2} - 1}))$

Step 3.5

Multiply the exponents in ${(t^{\frac{3}{2}})}^{- 2}$ .

Tap for more steps...

Step 3.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{\frac{3}{2} \cdot - 2} (\frac{3}{2} t^{\frac{3}{2} - 1}))$

Step 3.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.2.1

Factor $2$ out of $- 2$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} t^{\frac{3}{2} - 1}))$

Step 3.5.2.2

Cancel the common factor.

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} t^{\frac{3}{2} - 1}))$

Step 3.5.2.3

Rewrite the expression.

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{3 \cdot - 1} (\frac{3}{2} t^{\frac{3}{2} - 1}))$

Step 3.5.3

Multiply $3$ by $- 1$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} (\frac{3}{2} t^{\frac{3}{2} - 1}))$

Step 3.6

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} (\frac{3}{2} t^{\frac{3}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.7

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} (\frac{3}{2} t^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.8

Combine the numerators over the common denominator.

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} (\frac{3}{2} t^{\frac{3 - 1 \cdot 2}{2}}))$

Step 3.9

Simplify the numerator.

Tap for more steps...

Step 3.9.1

Multiply $- 1$ by $2$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} (\frac{3}{2} t^{\frac{3 - 2}{2}}))$

Step 3.9.2

Subtract $2$ from $3$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} (\frac{3}{2} t^{\frac{1}{2}}))$

Step 3.10

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- t^{- 3} \frac{3 t^{\frac{1}{2}}}{2})$

Step 3.11

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{\frac{1}{2}} t^{- 3}}{2})$

Step 3.12

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

Tap for more steps...

Step 3.12.1

Move $t^{- 3}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 (t^{- 3} t^{\frac{1}{2}})}{2})$

Step 3.12.2

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

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{- 3 + \frac{1}{2}}}{2})$

Step 3.12.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{- 3 \cdot \frac{2}{2} + \frac{1}{2}}}{2})$

Step 3.12.4

Combine $- 3$ and $\frac{2}{2}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{\frac{- 3 \cdot 2}{2} + \frac{1}{2}}}{2})$

Step 3.12.5

Combine the numerators over the common denominator.

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{\frac{- 3 \cdot 2 + 1}{2}}}{2})$

Step 3.12.6

Simplify the numerator.

Tap for more steps...

Step 3.12.6.1

Multiply $- 3$ by $2$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{\frac{- 6 + 1}{2}}}{2})$

Step 3.12.6.2

Add $- 6$ and $1$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{\frac{- 5}{2}}}{2})$

Step 3.12.7

Move the negative in front of the fraction.

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3 t^{- \frac{5}{2}}}{2})$

Step 3.13

Move $t^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{t^{\frac{6}{5}}} - 8 (- \frac{3}{2 t^{\frac{5}{2}}})$

Step 3.14

Multiply $- 1$ by $- 8$ .

$- \frac{1}{t^{\frac{6}{5}}} + 8 \frac{3}{2 t^{\frac{5}{2}}}$

Step 3.15

Combine $8$ and $\frac{3}{2 t^{\frac{5}{2}}}$ .

$- \frac{1}{t^{\frac{6}{5}}} + \frac{8 \cdot 3}{2 t^{\frac{5}{2}}}$

Step 3.16

Multiply $8$ by $3$ .

$- \frac{1}{t^{\frac{6}{5}}} + \frac{24}{2 t^{\frac{5}{2}}}$

Step 3.17

Factor $2$ out of $24$ .

$- \frac{1}{t^{\frac{6}{5}}} + \frac{2 \cdot 12}{2 t^{\frac{5}{2}}}$

Step 3.18

Cancel the common factors.

Tap for more steps...

Step 3.18.1

Factor $2$ out of $2 t^{\frac{5}{2}}$ .

$- \frac{1}{t^{\frac{6}{5}}} + \frac{2 \cdot 12}{2 (t^{\frac{5}{2}})}$

Step 3.18.2

Cancel the common factor.

$- \frac{1}{t^{\frac{6}{5}}} + \frac{2 \cdot 12}{2 t^{\frac{5}{2}}}$

Step 3.18.3

Rewrite the expression.

$- \frac{1}{t^{\frac{6}{5}}} + \frac{12}{t^{\frac{5}{2}}}$

Step 4

Reorder terms.

$\frac{12}{t^{\frac{5}{2}}} - \frac{1}{t^{\frac{6}{5}}}$