Calculus Examples

$\frac{e^{\frac{u}{10}}}{u^{3}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d u} [\frac{f (u)}{g (u)}]$ is $\frac{g (u) \frac{d}{d u} [f (u)] - f (u) \frac{d}{d u} [g (u)]}{{g (u)}^{2}}$ where $f (u) = e^{\frac{u}{10}}$ and $g (u) = u^{3}$ .

$\frac{u^{3} \frac{d}{d u} [e^{\frac{u}{10}}] - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{{(u^{3})}^{2}}$

Step 2

Multiply the exponents in ${(u^{3})}^{2}$ .

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Step 2.1

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

$\frac{u^{3} \frac{d}{d u} [e^{\frac{u}{10}}] - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{3 \cdot 2}}$

Step 2.2

Multiply $3$ by $2$ .

$\frac{u^{3} \frac{d}{d u} [e^{\frac{u}{10}}] - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = e^{u}$ and $g (u) = \frac{u}{10}$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $\frac{u}{10}$ .

$\frac{u^{3} (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d u} [\frac{u}{10}]) - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{u^{3} (e^{u_{1}} \frac{d}{d u} [\frac{u}{10}]) - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 3.3

Replace all occurrences of $u_{1}$ with $\frac{u}{10}$ .

$\frac{u^{3} (e^{\frac{u}{10}} \frac{d}{d u} [\frac{u}{10}]) - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 4

Differentiate.

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Step 4.1

Since $\frac{1}{10}$ is constant with respect to $u$ , the derivative of $\frac{u}{10}$ with respect to $u$ is $\frac{1}{10} \frac{d}{d u} [u]$ .

$\frac{u^{3} (e^{\frac{u}{10}} (\frac{1}{10} \frac{d}{d u} [u])) - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine $\frac{1}{10}$ and $e^{\frac{u}{10}}$ .

$\frac{u^{3} (\frac{e^{\frac{u}{10}}}{10} \frac{d}{d u} [u]) - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 4.2.2

Combine $\frac{e^{\frac{u}{10}}}{10}$ and $u^{3}$ .

$\frac{\frac{e^{\frac{u}{10}} u^{3}}{10} \frac{d}{d u} [u] - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 4.3

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

$\frac{\frac{e^{\frac{u}{10}} u^{3}}{10} \cdot 1 - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 4.4

Multiply $\frac{e^{\frac{u}{10}} u^{3}}{10}$ by $1$ .

$\frac{\frac{e^{\frac{u}{10}} u^{3}}{10} - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 5

Multiply $\frac{\frac{e^{\frac{u}{10}} u^{3}}{10} - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$ by $\frac{10}{10}$ .

$\frac{10}{10} \cdot \frac{\frac{e^{\frac{u}{10}} u^{3}}{10} - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}]}{u^{6}}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{10 (\frac{e^{\frac{u}{10}} u^{3}}{10} - e^{\frac{u}{10}} \frac{d}{d u} [u^{3}])}{10 u^{6}}$

Step 6.2

Apply the distributive property.

$\frac{10 \frac{e^{\frac{u}{10}} u^{3}}{10} + 10 (- e^{\frac{u}{10}} \frac{d}{d u} [u^{3}])}{10 u^{6}}$

Step 6.3

Cancel the common factor of $10$ .

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Step 6.3.1

Cancel the common factor.

$\frac{10 \frac{e^{\frac{u}{10}} u^{3}}{10} + 10 (- e^{\frac{u}{10}} \frac{d}{d u} [u^{3}])}{10 u^{6}}$

Step 6.3.2

Rewrite the expression.

$\frac{e^{\frac{u}{10}} u^{3} + 10 (- e^{\frac{u}{10}} \frac{d}{d u} [u^{3}])}{10 u^{6}}$

Step 6.4

Multiply $- 1$ by $10$ .

$\frac{e^{\frac{u}{10}} u^{3} - 10 (e^{\frac{u}{10}} \frac{d}{d u} [u^{3}])}{10 u^{6}}$

Step 7

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

$\frac{e^{\frac{u}{10}} u^{3} - 10 e^{\frac{u}{10}} (3 u^{2})}{10 u^{6}}$

Step 8

Multiply $3$ by $- 10$ .

$\frac{e^{\frac{u}{10}} u^{3} - 30 e^{\frac{u}{10}} u^{2}}{10 u^{6}}$

Step 9

Simplify.

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Step 9.1

Reorder factors in $e^{\frac{u}{10}} u^{3} - 30 e^{\frac{u}{10}} u^{2}$ .

$\frac{u^{3} e^{\frac{u}{10}} - 30 u^{2} e^{\frac{u}{10}}}{10 u^{6}}$

Step 9.2

Reorder terms.

$\frac{e^{\frac{u}{10}} u^{3} - 30 e^{\frac{u}{10}} u^{2}}{10 u^{6}}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

Factor $u^{2}$ out of $e^{\frac{u}{10}} u^{3} - 30 e^{\frac{u}{10}} u^{2}$ .

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Step 9.3.1.1

Factor $u^{2}$ out of $e^{\frac{u}{10}} u^{3}$ .

$\frac{u^{2} (e^{\frac{u}{10}} u) - 30 e^{\frac{u}{10}} u^{2}}{10 u^{6}}$

Step 9.3.1.2

Factor $u^{2}$ out of $- 30 e^{\frac{u}{10}} u^{2}$ .

$\frac{u^{2} (e^{\frac{u}{10}} u) + u^{2} (- 30 e^{\frac{u}{10}})}{10 u^{6}}$

Step 9.3.1.3

Factor $u^{2}$ out of $u^{2} (e^{\frac{u}{10}} u) + u^{2} (- 30 e^{\frac{u}{10}})$ .

$\frac{u^{2} (e^{\frac{u}{10}} u - 30 e^{\frac{u}{10}})}{10 u^{6}}$

Step 9.3.2

Factor $e^{\frac{u}{10}}$ out of $e^{\frac{u}{10}} u - 30 e^{\frac{u}{10}}$ .

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Step 9.3.2.1

Reorder $e^{\frac{u}{10}}$ and $u$ .

$\frac{u^{2} (u e^{\frac{u}{10}} - 30 e^{\frac{u}{10}})}{10 u^{6}}$

Step 9.3.2.2

Factor $e^{\frac{u}{10}}$ out of $u e^{\frac{u}{10}}$ .

$\frac{u^{2} (e^{\frac{u}{10}} u - 30 e^{\frac{u}{10}})}{10 u^{6}}$

Step 9.3.2.3

Factor $e^{\frac{u}{10}}$ out of $- 30 e^{\frac{u}{10}}$ .

$\frac{u^{2} (e^{\frac{u}{10}} u + e^{\frac{u}{10}} \cdot - 30)}{10 u^{6}}$

Step 9.3.2.4

Factor $e^{\frac{u}{10}}$ out of $e^{\frac{u}{10}} u + e^{\frac{u}{10}} \cdot - 30$ .

$\frac{u^{2} (e^{\frac{u}{10}} (u - 30))}{10 u^{6}}$

$\frac{u^{2} e^{\frac{u}{10}} (u - 30)}{10 u^{6}}$

Step 9.4

Cancel the common factor of $u^{2}$ and $u^{6}$ .

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Step 9.4.1

Factor $u^{2}$ out of $u^{2} e^{\frac{u}{10}} (u - 30)$ .

$\frac{u^{2} (e^{\frac{u}{10}} (u - 30))}{10 u^{6}}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $u^{2}$ out of $10 u^{6}$ .

$\frac{u^{2} (e^{\frac{u}{10}} (u - 30))}{u^{2} (10 u^{4})}$

Step 9.4.2.2

Cancel the common factor.

$\frac{u^{2} (e^{\frac{u}{10}} (u - 30))}{u^{2} (10 u^{4})}$

Step 9.4.2.3

Rewrite the expression.

$\frac{e^{\frac{u}{10}} (u - 30)}{10 u^{4}}$