Calculus Examples

$h (y) = y^{- 4} - 9 y + 8 y^{- 2} + 12$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $y^{- 4} - 9 y + 8 y^{- 2} + 12$ with respect to $y$ is $\frac{d}{d y} [y^{- 4}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [8 y^{- 2}] + \frac{d}{d y} [12]$ .

$\frac{d}{d y} [y^{- 4}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [8 y^{- 2}] + \frac{d}{d y} [12]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = - 4$ .

$- 4 y^{- 5} + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [8 y^{- 2}] + \frac{d}{d y} [12]$

Step 2

Evaluate $\frac{d}{d y} [- 9 y]$ .

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Step 2.1

Since $- 9$ is constant with respect to $y$ , the derivative of $- 9 y$ with respect to $y$ is $- 9 \frac{d}{d y} [y]$ .

$- 4 y^{- 5} - 9 \frac{d}{d y} [y] + \frac{d}{d y} [8 y^{- 2}] + \frac{d}{d y} [12]$

Step 2.2

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

$- 4 y^{- 5} - 9 \cdot 1 + \frac{d}{d y} [8 y^{- 2}] + \frac{d}{d y} [12]$

Step 2.3

Multiply $- 9$ by $1$ .

$- 4 y^{- 5} - 9 + \frac{d}{d y} [8 y^{- 2}] + \frac{d}{d y} [12]$

Step 3

Evaluate $\frac{d}{d y} [8 y^{- 2}]$ .

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Step 3.1

Since $8$ is constant with respect to $y$ , the derivative of $8 y^{- 2}$ with respect to $y$ is $8 \frac{d}{d y} [y^{- 2}]$ .

$- 4 y^{- 5} - 9 + 8 \frac{d}{d y} [y^{- 2}] + \frac{d}{d y} [12]$

Step 3.2

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

$- 4 y^{- 5} - 9 + 8 (- 2 y^{- 3}) + \frac{d}{d y} [12]$

Step 3.3

Multiply $- 2$ by $8$ .

$- 4 y^{- 5} - 9 - 16 y^{- 3} + \frac{d}{d y} [12]$

Step 4

Since $12$ is constant with respect to $y$ , the derivative of $12$ with respect to $y$ is $0$ .

$- 4 y^{- 5} - 9 - 16 y^{- 3} + 0$

Step 5

Simplify.

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Step 5.1

Add $- 4 y^{- 5} - 9 - 16 y^{- 3}$ and $0$ .

$- 4 y^{- 5} - 9 - 16 y^{- 3}$

Step 5.2

Reorder terms.

$- 16 y^{- 3} - 4 y^{- 5} - 9$

Step 5.3

Simplify each term.

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Step 5.3.1

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

$- 16 \frac{1}{y^{3}} - 4 y^{- 5} - 9$

Step 5.3.2

Combine $- 16$ and $\frac{1}{y^{3}}$ .

$\frac{- 16}{y^{3}} - 4 y^{- 5} - 9$

Step 5.3.3

Move the negative in front of the fraction.

$- \frac{16}{y^{3}} - 4 y^{- 5} - 9$

Step 5.3.4

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

$- \frac{16}{y^{3}} - 4 \frac{1}{y^{5}} - 9$

Step 5.3.5

Combine $- 4$ and $\frac{1}{y^{5}}$ .

$- \frac{16}{y^{3}} + \frac{- 4}{y^{5}} - 9$

Step 5.3.6

Move the negative in front of the fraction.

$- \frac{16}{y^{3}} - \frac{4}{y^{5}} - 9$