Calculus Examples

$g (y) = \frac{{(y - 1)}^{3}}{{(y^{2} + 2 y)}^{9}}$

Step 1

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 $f (y) = {(y - 1)}^{3}$ and $g (y) = {(y^{2} + 2 y)}^{9}$ .

$\frac{{(y^{2} + 2 y)}^{9} \frac{d}{d y} [{(y - 1)}^{3}] - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{({(y^{2} + 2 y)}^{9})}^{2}}$

Step 2

Multiply the exponents in ${({(y^{2} + 2 y)}^{9})}^{2}$ .

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Step 2.1

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

$\frac{{(y^{2} + 2 y)}^{9} \frac{d}{d y} [{(y - 1)}^{3}] - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{9 \cdot 2}}$

Step 2.2

Multiply $9$ by $2$ .

$\frac{{(y^{2} + 2 y)}^{9} \frac{d}{d y} [{(y - 1)}^{3}] - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $y - 1$ .

$\frac{{(y^{2} + 2 y)}^{9} (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d y} [y - 1]) - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 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 $n = 3$ .

$\frac{{(y^{2} + 2 y)}^{9} (3 {u_{1}}^{2} \frac{d}{d y} [y - 1]) - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 3.3

Replace all occurrences of $u_{1}$ with $y - 1$ .

$\frac{{(y^{2} + 2 y)}^{9} (3 {(y - 1)}^{2} \frac{d}{d y} [y - 1]) - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 4

Differentiate.

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Step 4.1

Move $3$ to the left of ${(y^{2} + 2 y)}^{9}$ .

$\frac{3 \cdot {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} \frac{d}{d y} [y - 1] - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 4.2

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} (\frac{d}{d y} [y] + \frac{d}{d y} [- 1]) - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 4.3

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} (1 + \frac{d}{d y} [- 1]) - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 4.4

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} (1 + 0) - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $1$ and $0$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} \cdot 1 - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 4.5.2

Multiply $3$ by $1$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - {(y - 1)}^{3} \frac{d}{d y} [{(y^{2} + 2 y)}^{9}]}{{(y^{2} + 2 y)}^{18}}$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{9}$ and $g (y) = y^{2} + 2 y$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $y^{2} + 2 y$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - {(y - 1)}^{3} (\frac{d}{d u_{2}} [{u_{2}}^{9}] \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{18}}$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 9$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - {(y - 1)}^{3} (9 {u_{2}}^{8} \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{18}}$

Step 5.3

Replace all occurrences of $u_{2}$ with $y^{2} + 2 y$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - {(y - 1)}^{3} (9 {(y^{2} + 2 y)}^{8} \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{18}}$

Step 6

Differentiate.

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Step 6.1

Multiply $9$ by $- 1$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{18}}$

Step 6.2

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (\frac{d}{d y} [y^{2}] + \frac{d}{d y} [2 y]))}{{(y^{2} + 2 y)}^{18}}$

Step 6.3

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + \frac{d}{d y} [2 y]))}{{(y^{2} + 2 y)}^{18}}$

Step 6.4

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + 2 \frac{d}{d y} [y]))}{{(y^{2} + 2 y)}^{18}}$

Step 6.5

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

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + 2 \cdot 1))}{{(y^{2} + 2 y)}^{18}}$

Step 6.6

Multiply $2$ by $1$ .

$\frac{3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor $3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2}$ out of $3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2} - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + 2))$ .

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Step 7.1.1.1

Factor $3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2}$ out of $3 {(y^{2} + 2 y)}^{9} {(y - 1)}^{2}$ .

$\frac{3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2} (y^{2} + 2 y) - 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.1.2

Factor $3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2}$ out of $- 9 {(y - 1)}^{3} ({(y^{2} + 2 y)}^{8} (2 y + 2))$ .

$\frac{3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2} (y^{2} + 2 y) + 3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2} (- 3 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.1.3

Factor $3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2}$ out of $3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2} (y^{2} + 2 y) + 3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2} (- 3 (y - 1) (2 y + 2))$ .

$\frac{3 {(y^{2} + 2 y)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.2

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

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Step 7.1.2.1

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

$\frac{3 {(y \cdot y + 2 y)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.2.2

Factor $y$ out of $2 y$ .

$\frac{3 {(y \cdot y + y \cdot 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.2.3

Factor $y$ out of $y \cdot y + y \cdot 2$ .

$\frac{3 {(y (y + 2))}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.3

Apply the product rule to $y (y + 2)$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4

Simplify each term.

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Step 7.1.4.1

Apply the distributive property.

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y + (- 3 y - 3 \cdot - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.2

Multiply $- 3$ by $- 1$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y + (- 3 y + 3) (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.3

Expand $(- 3 y + 3) (2 y + 2)$ using the FOIL Method.

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Step 7.1.4.3.1

Apply the distributive property.

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 y (2 y + 2) + 3 (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.3.2

Apply the distributive property.

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 y (2 y) - 3 y \cdot 2 + 3 (2 y + 2))}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.3.3

Apply the distributive property.

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 y (2 y) - 3 y \cdot 2 + 3 (2 y) + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4

Simplify and combine like terms.

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Step 7.1.4.4.1

Simplify each term.

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Step 7.1.4.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 \cdot 2 y \cdot y - 3 y \cdot 2 + 3 (2 y) + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 7.1.4.4.1.2.1

Move $y$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 \cdot 2 (y \cdot y) - 3 y \cdot 2 + 3 (2 y) + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.1.2.2

Multiply $y$ by $y$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 3 \cdot 2 y^{2} - 3 y \cdot 2 + 3 (2 y) + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.1.3

Multiply $- 3$ by $2$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 6 y^{2} - 3 y \cdot 2 + 3 (2 y) + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.1.4

Multiply $2$ by $- 3$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 6 y^{2} - 6 y + 3 (2 y) + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.1.5

Multiply $2$ by $3$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 6 y^{2} - 6 y + 6 y + 3 \cdot 2)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.1.6

Multiply $3$ by $2$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 6 y^{2} - 6 y + 6 y + 6)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.2

Add $- 6 y$ and $6 y$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 6 y^{2} + 0 + 6)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.4.4.3

Add $- 6 y^{2}$ and $0$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (y^{2} + 2 y - 6 y^{2} + 6)}{{(y^{2} + 2 y)}^{18}}$

Step 7.1.5

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

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{{(y^{2} + 2 y)}^{18}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

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

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Step 7.2.1.1

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

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{{(y \cdot y + 2 y)}^{18}}$

Step 7.2.1.2

Factor $y$ out of $2 y$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{{(y \cdot y + y \cdot 2)}^{18}}$

Step 7.2.1.3

Factor $y$ out of $y \cdot y + y \cdot 2$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{{(y (y + 2))}^{18}}$

Step 7.2.2

Apply the product rule to $y (y + 2)$ .

$\frac{3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{y^{18} {(y + 2)}^{18}}$

Step 7.3

Cancel the common factor of $y^{8}$ and $y^{18}$ .

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Step 7.3.1

Factor $y^{8}$ out of $3 y^{8} {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)$ .

$\frac{y^{8} (3 {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6))}{y^{18} {(y + 2)}^{18}}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $y^{8}$ out of $y^{18} {(y + 2)}^{18}$ .

$\frac{y^{8} (3 {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6))}{y^{8} (y^{10} {(y + 2)}^{18})}$

Step 7.3.2.2

Cancel the common factor.

$\frac{y^{8} (3 {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6))}{y^{8} (y^{10} {(y + 2)}^{18})}$

Step 7.3.2.3

Rewrite the expression.

$\frac{3 {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{y^{10} {(y + 2)}^{18}}$

Step 7.4

Cancel the common factor of ${(y + 2)}^{8}$ and ${(y + 2)}^{18}$ .

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Step 7.4.1

Factor ${(y + 2)}^{8}$ out of $3 {(y + 2)}^{8} {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)$ .

$\frac{{(y + 2)}^{8} (3 {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6))}{y^{10} {(y + 2)}^{18}}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor ${(y + 2)}^{8}$ out of $y^{10} {(y + 2)}^{18}$ .

$\frac{{(y + 2)}^{8} (3 {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6))}{{(y + 2)}^{8} (y^{10} {(y + 2)}^{10})}$

Step 7.4.2.2

Cancel the common factor.

$\frac{{(y + 2)}^{8} (3 {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6))}{{(y + 2)}^{8} (y^{10} {(y + 2)}^{10})}$

Step 7.4.2.3

Rewrite the expression.

$\frac{3 {(y - 1)}^{2} (- 5 y^{2} + 2 y + 6)}{y^{10} {(y + 2)}^{10}}$

Step 7.5

Factor $- 1$ out of $- 5 y^{2}$ .

$\frac{3 {(y - 1)}^{2} (- (5 y^{2}) + 2 y + 6)}{y^{10} {(y + 2)}^{10}}$

Step 7.6

Factor $- 1$ out of $2 y$ .

$\frac{3 {(y - 1)}^{2} (- (5 y^{2}) - (- 2 y) + 6)}{y^{10} {(y + 2)}^{10}}$

Step 7.7

Factor $- 1$ out of $- (5 y^{2}) - (- 2 y)$ .

$\frac{3 {(y - 1)}^{2} (- (5 y^{2} - 2 y) + 6)}{y^{10} {(y + 2)}^{10}}$

Step 7.8

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{3 {(y - 1)}^{2} (- (5 y^{2} - 2 y) - 1 (- 6))}{y^{10} {(y + 2)}^{10}}$

Step 7.9

Factor $- 1$ out of $- (5 y^{2} - 2 y) - 1 (- 6)$ .

$\frac{3 {(y - 1)}^{2} (- (5 y^{2} - 2 y - 6))}{y^{10} {(y + 2)}^{10}}$

Step 7.10

Rewrite $- (5 y^{2} - 2 y - 6)$ as $- 1 (5 y^{2} - 2 y - 6)$ .

$\frac{3 {(y - 1)}^{2} (- 1 (5 y^{2} - 2 y - 6))}{y^{10} {(y + 2)}^{10}}$

Step 7.11

Move the negative in front of the fraction.

$- \frac{(3 {(y - 1)}^{2}) (5 y^{2} - 2 y - 6)}{y^{10} {(y + 2)}^{10}}$

Step 7.12

Reorder factors in $- \frac{(3 {(y - 1)}^{2}) (5 y^{2} - 2 y - 6)}{y^{10} {(y + 2)}^{10}}$ .

$- \frac{3 {(y - 1)}^{2} (5 y^{2} - 2 y - 6)}{y^{10} {(y + 2)}^{10}}$