Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{8}{t^{\frac{1}{6}}} - \frac{5}{t^{\frac{2}{5}}}$ with respect to $t$ is $\frac{d}{d t} [\frac{8}{t^{\frac{1}{6}}}] + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$ .

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

Step 2

Evaluate $\frac{d}{d t} [\frac{8}{t^{\frac{1}{6}}}]$ .

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Step 2.1

Since $8$ is constant with respect to $t$ , the derivative of $\frac{8}{t^{\frac{1}{6}}}$ with respect to $t$ is $8 \frac{d}{d t} [\frac{1}{t^{\frac{1}{6}}}]$ .

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

Step 2.2

Rewrite $\frac{1}{t^{\frac{1}{6}}}$ as ${(t^{\frac{1}{6}})}^{- 1}$ .

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

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 $g (t) = t^{\frac{1}{6}}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $t^{\frac{1}{6}}$ .

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

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 $n = - 1$ .

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

Step 2.3.3

Replace all occurrences of $u_{1}$ with $t^{\frac{1}{6}}$ .

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

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{6}$ .

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

Step 2.5

Multiply the exponents in ${(t^{\frac{1}{6}})}^{- 2}$ .

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Step 2.5.1

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

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

Step 2.5.2

Cancel the common factor of $2$ .

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Step 2.5.2.1

Factor $2$ out of $6$ .

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

Step 2.5.2.2

Factor $2$ out of $- 2$ .

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

Step 2.5.2.3

Cancel the common factor.

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

Step 2.5.2.4

Rewrite the expression.

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

Step 2.5.3

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

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

Step 2.5.4

Move the negative in front of the fraction.

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

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 2.7

Combine $- 1$ and $\frac{6}{6}$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $6$ .

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

Step 2.9.2

Subtract $6$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

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

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

Step 2.12

Combine $\frac{t^{- \frac{5}{6}}}{6}$ and $t^{- \frac{1}{3}}$ .

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

Step 2.13

Multiply $t^{- \frac{5}{6}}$ by $t^{- \frac{1}{3}}$ by adding the exponents.

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Step 2.13.1

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

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

Step 2.13.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.13.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.13.3.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

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

Step 2.13.3.2

Multiply $3$ by $2$ .

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

Step 2.13.4

Combine the numerators over the common denominator.

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

Step 2.13.5

Subtract $2$ from $- 5$ .

$8 (- \frac{t^{\frac{- 7}{6}}}{6}) + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.13.6

Move the negative in front of the fraction.

$8 (- \frac{t^{- \frac{7}{6}}}{6}) + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.14

Move $t^{- \frac{7}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 (- \frac{1}{6 t^{\frac{7}{6}}}) + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.15

Multiply $- 1$ by $8$ .

$- 8 \frac{1}{6 t^{\frac{7}{6}}} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.16

Combine $- 8$ and $\frac{1}{6 t^{\frac{7}{6}}}$ .

$\frac{- 8}{6 t^{\frac{7}{6}}} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.17

Factor $2$ out of $- 8$ .

$\frac{2 (- 4)}{6 t^{\frac{7}{6}}} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.18

Cancel the common factors.

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Step 2.18.1

Factor $2$ out of $6 t^{\frac{7}{6}}$ .

$\frac{2 (- 4)}{2 (3 t^{\frac{7}{6}})} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.18.2

Cancel the common factor.

$\frac{2 \cdot - 4}{2 (3 t^{\frac{7}{6}})} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.18.3

Rewrite the expression.

$\frac{- 4}{3 t^{\frac{7}{6}}} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 2.19

Move the negative in front of the fraction.

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$

Step 3

Evaluate $\frac{d}{d t} [- \frac{5}{t^{\frac{2}{5}}}]$ .

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Step 3.1

Since $- 5$ is constant with respect to $t$ , the derivative of $- \frac{5}{t^{\frac{2}{5}}}$ with respect to $t$ is $- 5 \frac{d}{d t} [\frac{1}{t^{\frac{2}{5}}}]$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 \frac{d}{d t} [\frac{1}{t^{\frac{2}{5}}}]$

Step 3.2

Rewrite $\frac{1}{t^{\frac{2}{5}}}$ as ${(t^{\frac{2}{5}})}^{- 1}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 \frac{d}{d t} [{(t^{\frac{2}{5}})}^{- 1}]$

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 $g (t) = t^{\frac{2}{5}}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $t^{\frac{2}{5}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d t} [t^{\frac{2}{5}}])$

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 $n = - 1$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- {u_{2}}^{- 2} \frac{d}{d t} [t^{\frac{2}{5}}])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $t^{\frac{2}{5}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- {(t^{\frac{2}{5}})}^{- 2} \frac{d}{d t} [t^{\frac{2}{5}}])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{2}{5}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- {(t^{\frac{2}{5}})}^{- 2} (\frac{2}{5} t^{\frac{2}{5} - 1}))$

Step 3.5

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

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Step 3.5.1

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

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{\frac{2}{5} \cdot - 2} (\frac{2}{5} t^{\frac{2}{5} - 1}))$

Step 3.5.2

Multiply $\frac{2}{5} \cdot - 2$ .

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Step 3.5.2.1

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

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{\frac{2 \cdot - 2}{5}} (\frac{2}{5} t^{\frac{2}{5} - 1}))$

Step 3.5.2.2

Multiply $2$ by $- 2$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{\frac{- 4}{5}} (\frac{2}{5} t^{\frac{2}{5} - 1}))$

Step 3.5.3

Move the negative in front of the fraction.

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{\frac{2}{5} - 1}))$

Step 3.6

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

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{\frac{2}{5} - 1 \cdot \frac{5}{5}}))$

Step 3.7

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

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}}))$

Step 3.8

Combine the numerators over the common denominator.

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{\frac{2 - 1 \cdot 5}{5}}))$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $- 1$ by $5$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{\frac{2 - 5}{5}}))$

Step 3.9.2

Subtract $5$ from $2$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{\frac{- 3}{5}}))$

Step 3.10

Move the negative in front of the fraction.

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} (\frac{2}{5} t^{- \frac{3}{5}}))$

Step 3.11

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

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- t^{- \frac{4}{5}} \frac{2 t^{- \frac{3}{5}}}{5})$

Step 3.12

Combine $\frac{2 t^{- \frac{3}{5}}}{5}$ and $t^{- \frac{4}{5}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2 t^{- \frac{3}{5}} t^{- \frac{4}{5}}}{5})$

Step 3.13

Multiply $t^{- \frac{3}{5}}$ by $t^{- \frac{4}{5}}$ by adding the exponents.

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Step 3.13.1

Move $t^{- \frac{4}{5}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2 (t^{- \frac{4}{5}} t^{- \frac{3}{5}})}{5})$

Step 3.13.2

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

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2 t^{- \frac{4}{5} - \frac{3}{5}}}{5})$

Step 3.13.3

Combine the numerators over the common denominator.

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2 t^{\frac{- 4 - 3}{5}}}{5})$

Step 3.13.4

Subtract $3$ from $- 4$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2 t^{\frac{- 7}{5}}}{5})$

Step 3.13.5

Move the negative in front of the fraction.

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2 t^{- \frac{7}{5}}}{5})$

Step 3.14

Move $t^{- \frac{7}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} - 5 (- \frac{2}{5 t^{\frac{7}{5}}})$

Step 3.15

Multiply $- 1$ by $- 5$ .

$- \frac{4}{3 t^{\frac{7}{6}}} + 5 \frac{2}{5 t^{\frac{7}{5}}}$

Step 3.16

Combine $5$ and $\frac{2}{5 t^{\frac{7}{5}}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{5 \cdot 2}{5 t^{\frac{7}{5}}}$

Step 3.17

Multiply $5$ by $2$ .

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{10}{5 t^{\frac{7}{5}}}$

Step 3.18

Factor $5$ out of $10$ .

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{5 \cdot 2}{5 t^{\frac{7}{5}}}$

Step 3.19

Cancel the common factors.

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Step 3.19.1

Factor $5$ out of $5 t^{\frac{7}{5}}$ .

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{5 \cdot 2}{5 (t^{\frac{7}{5}})}$

Step 3.19.2

Cancel the common factor.

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{5 \cdot 2}{5 t^{\frac{7}{5}}}$

Step 3.19.3

Rewrite the expression.

$- \frac{4}{3 t^{\frac{7}{6}}} + \frac{2}{t^{\frac{7}{5}}}$

Step 4

Reorder terms.

$\frac{2}{t^{\frac{7}{5}}} - \frac{4}{3 t^{\frac{7}{6}}}$