Calculus Examples

$\frac{3}{y - 3}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $3$ is constant with respect to $y$ , the derivative of $\frac{3}{y - 3}$ with respect to $y$ is $3 \frac{d}{d y} [\frac{1}{y - 3}]$ .

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

Step 1.2

Rewrite $\frac{1}{y - 3}$ as ${(y - 3)}^{- 1}$ .

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

Step 2

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 $g (y) = y - 3$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $y - 3$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $y - 3$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $3$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

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

Step 3.5.2

Multiply $- 3$ by $1$ .

$- 3 {(y - 3)}^{- 2}$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Combine terms.

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Step 4.2.1

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

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

Step 4.2.2

Move the negative in front of the fraction.

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