Algebra Examples

$4 y {(8 y + 24)}^{- 1}$

Step 1

Since $4$ is constant with respect to $y$ , the derivative of $4 y {(8 y + 24)}^{- 1}$ with respect to $y$ is $4 \frac{d}{d y} [y {(8 y + 24)}^{- 1}]$ .

$4 \frac{d}{d y} [y {(8 y + 24)}^{- 1}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = {(8 y + 24)}^{- 1}$ .

$4 (y \frac{d}{d y} [{(8 y + 24)}^{- 1}] + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = 8 y + 24$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $8 y + 24$ .

$4 (y (\frac{d}{d u} [u^{- 1}] \frac{d}{d y} [8 y + 24]) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 3.2

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

$4 (y (- u^{- 2} \frac{d}{d y} [8 y + 24]) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 3.3

Replace all occurrences of $u$ with $8 y + 24$ .

$4 (y (- {(8 y + 24)}^{- 2} \frac{d}{d y} [8 y + 24]) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $8 y + 24$ with respect to $y$ is $\frac{d}{d y} [8 y] + \frac{d}{d y} [24]$ .

$4 (y (- {(8 y + 24)}^{- 2} (\frac{d}{d y} [8 y] + \frac{d}{d y} [24])) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.2

Since $8$ is constant with respect to $y$ , the derivative of $8 y$ with respect to $y$ is $8 \frac{d}{d y} [y]$ .

$4 (y (- {(8 y + 24)}^{- 2} (8 \frac{d}{d y} [y] + \frac{d}{d y} [24])) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.3

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

$4 (y (- {(8 y + 24)}^{- 2} (8 \cdot 1 + \frac{d}{d y} [24])) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.4

Multiply $8$ by $1$ .

$4 (y (- {(8 y + 24)}^{- 2} (8 + \frac{d}{d y} [24])) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.5

Since $24$ is constant with respect to $y$ , the derivative of $24$ with respect to $y$ is $0$ .

$4 (y (- {(8 y + 24)}^{- 2} (8 + 0)) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $8$ and $0$ .

$4 (y (- {(8 y + 24)}^{- 2} \cdot 8) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.6.2

Multiply $8$ by $- 1$ .

$4 (y (- 8 {(8 y + 24)}^{- 2}) + {(8 y + 24)}^{- 1} \frac{d}{d y} [y])$

Step 4.7

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

$4 (y (- 8 {(8 y + 24)}^{- 2}) + {(8 y + 24)}^{- 1} \cdot 1)$

Step 4.8

Multiply ${(8 y + 24)}^{- 1}$ by $1$ .

$4 (y (- 8 {(8 y + 24)}^{- 2}) + {(8 y + 24)}^{- 1})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 (y (- 8 {(8 y + 24)}^{- 2})) + 4 {(8 y + 24)}^{- 1}$

Step 5.2

Multiply $- 8$ by $4$ .

$- 32 y {(8 y + 24)}^{- 2} + 4 {(8 y + 24)}^{- 1}$

Step 5.3

Simplify each term.

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Step 5.3.1

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

$- 32 y \frac{1}{{(8 y + 24)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.2

Simplify the denominator.

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Step 5.3.2.1

Factor $8$ out of $8 y + 24$ .

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Step 5.3.2.1.1

Factor $8$ out of $8 y$ .

$- 32 y \frac{1}{{(8 (y) + 24)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.2.1.2

Factor $8$ out of $24$ .

$- 32 y \frac{1}{{(8 y + 8 \cdot 3)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.2.1.3

Factor $8$ out of $8 y + 8 \cdot 3$ .

$- 32 y \frac{1}{{(8 (y + 3))}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.2.2

Apply the product rule to $8 (y + 3)$ .

$- 32 y \frac{1}{8^{2} {(y + 3)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.2.3

Raise $8$ to the power of $2$ .

$- 32 y \frac{1}{64 {(y + 3)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.3

Cancel the common factor of $32$ .

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Step 5.3.3.1

Factor $32$ out of $- 32 y$ .

$32 (- y) \frac{1}{64 {(y + 3)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.3.2

Factor $32$ out of $64 {(y + 3)}^{2}$ .

$32 (- y) \frac{1}{32 (2 {(y + 3)}^{2})} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.3.3

Cancel the common factor.

$32 (- y) \frac{1}{32 (2 {(y + 3)}^{2})} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.3.4

Rewrite the expression.

$- y \frac{1}{2 {(y + 3)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.4

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

$- \frac{y}{2 {(y + 3)}^{2}} + 4 {(8 y + 24)}^{- 1}$

Step 5.3.5

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

$- \frac{y}{2 {(y + 3)}^{2}} + 4 \frac{1}{8 y + 24}$

Step 5.3.6

Factor $8$ out of $8 y + 24$ .

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Step 5.3.6.1

Factor $8$ out of $8 y$ .

$- \frac{y}{2 {(y + 3)}^{2}} + 4 \frac{1}{8 (y) + 24}$

Step 5.3.6.2

Factor $8$ out of $24$ .

$- \frac{y}{2 {(y + 3)}^{2}} + 4 \frac{1}{8 y + 8 \cdot 3}$

Step 5.3.6.3

Factor $8$ out of $8 y + 8 \cdot 3$ .

$- \frac{y}{2 {(y + 3)}^{2}} + 4 \frac{1}{8 (y + 3)}$

Step 5.3.7

Cancel the common factor of $4$ .

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Step 5.3.7.1

Factor $4$ out of $8 (y + 3)$ .

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

Step 5.3.7.2

Cancel the common factor.

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

Step 5.3.7.3

Rewrite the expression.

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