Calculus Examples

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

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 $g (y) = \frac{y + 2}{3 - y}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{y + 2}{3 - y}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{y + 2}{3 - y}$ .

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

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 $g (y) = 3 - y$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $3 - y$ by $1$ .

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

Step 3.5

By the Sum Rule, the derivative of $3 - y$ with respect to $y$ is $\frac{d}{d y} [3] + \frac{d}{d y} [- y]$ .

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

Step 3.6

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

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

Step 3.7

Add $0$ and $\frac{d}{d y} [- y]$ .

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

Step 3.8

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

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

Step 3.9

Multiply.

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Step 3.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.9.2

Multiply $y + 2$ by $1$ .

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

Step 3.10

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

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

Step 3.11

Simplify by adding terms.

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Step 3.11.1

Multiply $y + 2$ by $1$ .

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

Step 3.11.2

Add $- y$ and $y$ .

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

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Add $3$ and $0$ .

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

Step 3.11.3.2

Add $3$ and $2$ .

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

Step 3.11.3.3

Reorder terms.

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{y + 2}{3 - y}$ .

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

Step 4.2

Combine terms.

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Step 4.2.1

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

$\frac{3 {(y + 2)}^{2}}{{(3 - y)}^{2}} \cdot \frac{5}{{(- y + 3)}^{2}}$

Step 4.2.2

Multiply $\frac{3 {(y + 2)}^{2}}{{(3 - y)}^{2}}$ by $\frac{5}{{(- y + 3)}^{2}}$ .

$\frac{3 {(y + 2)}^{2} \cdot 5}{{(3 - y)}^{2} {(- y + 3)}^{2}}$

Step 4.2.3

Multiply $5$ by $3$ .

$\frac{15 {(y + 2)}^{2}}{{(3 - y)}^{2} {(- y + 3)}^{2}}$

Step 4.2.4

Reorder terms.

$\frac{15 {(y + 2)}^{2}}{{(- y + 3)}^{2} {(- y + 3)}^{2}}$

Step 4.2.5

Multiply ${(- y + 3)}^{2}$ by ${(- y + 3)}^{2}$ by adding the exponents.

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Step 4.2.5.1

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

$\frac{15 {(y + 2)}^{2}}{{(- y + 3)}^{2 + 2}}$

Step 4.2.5.2

Add $2$ and $2$ .

$\frac{15 {(y + 2)}^{2}}{{(- y + 3)}^{4}}$