Calculus Examples

$G (y) = {(\frac{y^{2}}{2 y - 1})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{3}$ and $g (y) = \frac{y^{2}}{2 y - 1}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{y^{2}}{2 y - 1}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d y} [\frac{y^{2}}{2 y - 1}]$

Step 1.2

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

$3 u^{2} \frac{d}{d y} [\frac{y^{2}}{2 y - 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{y^{2}}{2 y - 1}$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Move $2$ to the left of $2 y - 1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $2$ by $1$ .

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

Step 3.7

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

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

Step 3.8

Combine fractions.

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Step 3.8.1

Add $2$ and $0$ .

$3 {(\frac{y^{2}}{2 y - 1})}^{2} \frac{2 (2 y - 1) y - y^{2} \cdot 2}{{(2 y - 1)}^{2}}$

Step 3.8.2

Multiply $2$ by $- 1$ .

$3 {(\frac{y^{2}}{2 y - 1})}^{2} \frac{2 (2 y - 1) y - 2 y^{2}}{{(2 y - 1)}^{2}}$

Step 3.8.3

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

$\frac{(2 (2 y - 1) y - 2 y^{2}) \cdot 3}{{(2 y - 1)}^{2}} {(\frac{y^{2}}{2 y - 1})}^{2}$

Step 3.8.4

Move $3$ to the left of $2 (2 y - 1) y - 2 y^{2}$ .

$\frac{3 \cdot (2 (2 y - 1) y - 2 y^{2})}{{(2 y - 1)}^{2}} {(\frac{y^{2}}{2 y - 1})}^{2}$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{y^{2}}{2 y - 1}$ .

$\frac{3 (2 (2 y - 1) y - 2 y^{2})}{{(2 y - 1)}^{2}} \cdot \frac{{(y^{2})}^{2}}{{(2 y - 1)}^{2}}$

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Combine terms.

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Step 4.5.1

Multiply $2$ by $2$ .

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

Step 4.5.2

Raise $y$ to the power of $1$ .

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

Step 4.5.3

Raise $y$ to the power of $1$ .

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

Step 4.5.4

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

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

Step 4.5.5

Add $1$ and $1$ .

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

Step 4.5.6

Multiply $4$ by $3$ .

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

Step 4.5.7

Multiply $2$ by $- 1$ .

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

Step 4.5.8

Multiply $- 2$ by $3$ .

$\frac{12 y^{2} - 6 y + 3 (- 2 y^{2})}{{(2 y - 1)}^{2}} \cdot \frac{{(y^{2})}^{2}}{{(2 y - 1)}^{2}}$

Step 4.5.9

Multiply $- 2$ by $3$ .

$\frac{12 y^{2} - 6 y - 6 y^{2}}{{(2 y - 1)}^{2}} \cdot \frac{{(y^{2})}^{2}}{{(2 y - 1)}^{2}}$

Step 4.5.10

Subtract $6 y^{2}$ from $12 y^{2}$ .

$\frac{6 y^{2} - 6 y}{{(2 y - 1)}^{2}} \cdot \frac{{(y^{2})}^{2}}{{(2 y - 1)}^{2}}$

Step 4.5.11

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

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Step 4.5.11.1

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

$\frac{6 y^{2} - 6 y}{{(2 y - 1)}^{2}} \cdot \frac{y^{2 \cdot 2}}{{(2 y - 1)}^{2}}$

Step 4.5.11.2

Multiply $2$ by $2$ .

$\frac{6 y^{2} - 6 y}{{(2 y - 1)}^{2}} \cdot \frac{y^{4}}{{(2 y - 1)}^{2}}$

Step 4.5.12

Multiply $\frac{6 y^{2} - 6 y}{{(2 y - 1)}^{2}}$ by $\frac{y^{4}}{{(2 y - 1)}^{2}}$ .

$\frac{(6 y^{2} - 6 y) y^{4}}{{(2 y - 1)}^{2} {(2 y - 1)}^{2}}$

Step 4.5.13

Multiply ${(2 y - 1)}^{2}$ by ${(2 y - 1)}^{2}$ by adding the exponents.

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Step 4.5.13.1

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

$\frac{(6 y^{2} - 6 y) y^{4}}{{(2 y - 1)}^{2 + 2}}$

Step 4.5.13.2

Add $2$ and $2$ .

$\frac{(6 y^{2} - 6 y) y^{4}}{{(2 y - 1)}^{4}}$

Step 4.6

Reorder terms.

$\frac{y^{4} (6 y^{2} - 6 y)}{{(2 y - 1)}^{4}}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Factor $6 y$ out of $6 y^{2} - 6 y$ .

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Step 4.7.1.1

Factor $6 y$ out of $6 y^{2}$ .

$\frac{y^{4} (6 y (y) - 6 y)}{{(2 y - 1)}^{4}}$

Step 4.7.1.2

Factor $6 y$ out of $- 6 y$ .

$\frac{y^{4} (6 y (y) + 6 y (- 1))}{{(2 y - 1)}^{4}}$

Step 4.7.1.3

Factor $6 y$ out of $6 y (y) + 6 y (- 1)$ .

$\frac{y^{4} (6 y (y - 1))}{{(2 y - 1)}^{4}}$

$\frac{y^{4} \cdot 6 y (y - 1)}{{(2 y - 1)}^{4}}$

Step 4.7.2

Combine exponents.

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Step 4.7.2.1

Raise $y$ to the power of $1$ .

$\frac{6 (y^{4} y^{1}) (y - 1)}{{(2 y - 1)}^{4}}$

Step 4.7.2.2

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

$\frac{6 y^{4 + 1} (y - 1)}{{(2 y - 1)}^{4}}$

Step 4.7.2.3

Add $4$ and $1$ .

$\frac{6 y^{5} (y - 1)}{{(2 y - 1)}^{4}}$