Calculus Examples

$\ln (5 x^{2} + 10 y^{3}) \cdot (6 x^{- \frac{3}{2}})$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d y} [\ln (5 x^{2} + 10 y^{3}) \cdot (6 \frac{1}{x^{\frac{3}{2}}})]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Combine $6$ and $\frac{1}{x^{\frac{3}{2}}}$ .

$\frac{d}{d y} [\ln (5 x^{2} + 10 y^{3}) \cdot \frac{6}{x^{\frac{3}{2}}}]$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine $\ln (5 x^{2} + 10 y^{3})$ and $\frac{6}{x^{\frac{3}{2}}}$ .

$\frac{d}{d y} [\frac{\ln (5 x^{2} + 10 y^{3}) \cdot 6}{x^{\frac{3}{2}}}]$

Step 2.2.2

Move $6$ to the left of $\ln (5 x^{2} + 10 y^{3})$ .

$\frac{d}{d y} [\frac{6 \cdot \ln (5 x^{2} + 10 y^{3})}{x^{\frac{3}{2}}}]$

Step 2.3

Since $\frac{6}{x^{\frac{3}{2}}}$ is constant with respect to $y$ , the derivative of $\frac{6 \ln (5 x^{2} + 10 y^{3})}{x^{\frac{3}{2}}}$ with respect to $y$ is $\frac{6}{x^{\frac{3}{2}}} \frac{d}{d y} [\ln (5 x^{2} + 10 y^{3})]$ .

$\frac{6}{x^{\frac{3}{2}}} \frac{d}{d y} [\ln (5 x^{2} + 10 y^{3})]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \ln (y)$ and $g (y) = 5 x^{2} + 10 y^{3}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 10 y^{3}$ .

$\frac{6}{x^{\frac{3}{2}}} (\frac{d}{d u} [\ln (u)] \frac{d}{d y} [5 x^{2} + 10 y^{3}])$

Step 3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{6}{x^{\frac{3}{2}}} (\frac{1}{u} \frac{d}{d y} [5 x^{2} + 10 y^{3}])$

Step 3.3

Replace all occurrences of $u$ with $5 x^{2} + 10 y^{3}$ .

$\frac{6}{x^{\frac{3}{2}}} (\frac{1}{5 x^{2} + 10 y^{3}} \frac{d}{d y} [5 x^{2} + 10 y^{3}])$

Step 4

Differentiate.

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Step 4.1

Multiply $\frac{1}{5 x^{2} + 10 y^{3}}$ by $\frac{6}{x^{\frac{3}{2}}}$ .

$\frac{6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} \frac{d}{d y} [5 x^{2} + 10 y^{3}]$

Step 4.2

By the Sum Rule, the derivative of $5 x^{2} + 10 y^{3}$ with respect to $y$ is $\frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [10 y^{3}]$ .

$\frac{6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} (\frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [10 y^{3}])$

Step 4.3

Since $5 x^{2}$ is constant with respect to $y$ , the derivative of $5 x^{2}$ with respect to $y$ is $0$ .

$\frac{6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} (0 + \frac{d}{d y} [10 y^{3}])$

Step 4.4

Add $0$ and $\frac{d}{d y} [10 y^{3}]$ .

$\frac{6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} \frac{d}{d y} [10 y^{3}]$

Step 4.5

Since $10$ is constant with respect to $y$ , the derivative of $10 y^{3}$ with respect to $y$ is $10 \frac{d}{d y} [y^{3}]$ .

$\frac{6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} (10 \frac{d}{d y} [y^{3}])$

Step 4.6

Simplify terms.

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Step 4.6.1

Combine $10$ and $\frac{6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}}$ .

$\frac{10 \cdot 6}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} \frac{d}{d y} [y^{3}]$

Step 4.6.2

Multiply $10$ by $6$ .

$\frac{60}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} \frac{d}{d y} [y^{3}]$

Step 4.6.3

Factor $5$ out of $60$ .

$\frac{5 \cdot 12}{(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}} \frac{d}{d y} [y^{3}]$

Step 5

Cancel the common factors.

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Step 5.1

Factor $5$ out of $(5 x^{2} + 10 y^{3}) x^{\frac{3}{2}}$ .

$\frac{5 \cdot 12}{5 ((x^{2} + 2 y^{3}) x^{\frac{3}{2}})} \frac{d}{d y} [y^{3}]$

Step 5.2

Cancel the common factor.

$\frac{5 \cdot 12}{5 ((x^{2} + 2 y^{3}) x^{\frac{3}{2}})} \frac{d}{d y} [y^{3}]$

Step 5.3

Rewrite the expression.

$\frac{12}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}} \frac{d}{d y} [y^{3}]$

Step 6

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

$\frac{12}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}} (3 y^{2})$

Step 7

Combine fractions.

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Step 7.1

Combine $3$ and $\frac{12}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}}$ .

$\frac{3 \cdot 12}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}} y^{2}$

Step 7.2

Multiply $3$ by $12$ .

$\frac{36}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}} y^{2}$

Step 7.3

Combine $\frac{36}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}}$ and $y^{2}$ .

$\frac{36 y^{2}}{(x^{2} + 2 y^{3}) x^{\frac{3}{2}}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{36 y^{2}}{x^{2} x^{\frac{3}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.2

Multiply $x^{2}$ by $x^{\frac{3}{2}}$ by adding the exponents.

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Step 8.2.1

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

$\frac{36 y^{2}}{x^{2 + \frac{3}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.2.2

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

$\frac{36 y^{2}}{x^{2 \cdot \frac{2}{2} + \frac{3}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.2.3

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

$\frac{36 y^{2}}{x^{\frac{2 \cdot 2}{2} + \frac{3}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.2.4

Combine the numerators over the common denominator.

$\frac{36 y^{2}}{x^{\frac{2 \cdot 2 + 3}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.2.5

Simplify the numerator.

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Step 8.2.5.1

Multiply $2$ by $2$ .

$\frac{36 y^{2}}{x^{\frac{4 + 3}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.2.5.2

Add $4$ and $3$ .

$\frac{36 y^{2}}{x^{\frac{7}{2}} + 2 y^{3} x^{\frac{3}{2}}}$

Step 8.3

Reorder terms.

$\frac{36 y^{2}}{2 x^{\frac{3}{2}} y^{3} + x^{\frac{7}{2}}}$

Step 8.4

Simplify the denominator.

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Step 8.4.1

Factor $x^{\frac{3}{2}}$ out of $2 x^{\frac{3}{2}} y^{3} + x^{\frac{7}{2}}$ .

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Step 8.4.1.1

Factor $x^{\frac{3}{2}}$ out of $2 x^{\frac{3}{2}} y^{3}$ .

$\frac{36 y^{2}}{x^{\frac{3}{2}} (2 y^{3}) + x^{\frac{7}{2}}}$

Step 8.4.1.2

Factor $x^{\frac{3}{2}}$ out of $x^{\frac{7}{2}}$ .

$\frac{36 y^{2}}{x^{\frac{3}{2}} (2 y^{3}) + x^{\frac{3}{2}} x^{\frac{4}{2}}}$

Step 8.4.1.3

Factor $x^{\frac{3}{2}}$ out of $x^{\frac{3}{2}} (2 y^{3}) + x^{\frac{3}{2}} x^{\frac{4}{2}}$ .

$\frac{36 y^{2}}{x^{\frac{3}{2}} (2 y^{3} + x^{\frac{4}{2}})}$

Step 8.4.2

Divide $4$ by $2$ .

$\frac{36 y^{2}}{x^{\frac{3}{2}} (2 y^{3} + x^{2})}$