Calculus Examples

$y = \frac{4}{x^{2} - 3 x + 2}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{2} - 3 x + 2}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{2} - 3 x + 2}]$ .

$4 \frac{d}{d x} [\frac{1}{x^{2} - 3 x + 2}]$

Step 1.2

Rewrite $\frac{1}{x^{2} - 3 x + 2}$ as ${(x^{2} - 3 x + 2)}^{- 1}$ .

$4 \frac{d}{d x} [{(x^{2} - 3 x + 2)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 3 x + 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} - 3 x + 2$ .

$4 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 3 x + 2])$

Step 2.2

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

$4 (- u^{- 2} \frac{d}{d x} [x^{2} - 3 x + 2])$

Step 2.3

Replace all occurrences of $u$ with $x^{2} - 3 x + 2$ .

$4 (- {(x^{2} - 3 x + 2)}^{- 2} \frac{d}{d x} [x^{2} - 3 x + 2])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $4$ .

$- 4 ({(x^{2} - 3 x + 2)}^{- 2} \frac{d}{d x} [x^{2} - 3 x + 2])$

Step 3.2

By the Sum Rule, the derivative of $x^{2} - 3 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$ .

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2])$

Step 3.3

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

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2])$

Step 3.4

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

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 3.5

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

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (2 x - 3 \cdot 1 + \frac{d}{d x} [2])$

Step 3.6

Multiply $- 3$ by $1$ .

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (2 x - 3 + \frac{d}{d x} [2])$

Step 3.7

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

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (2 x - 3 + 0)$

Step 3.8

Add $2 x - 3$ and $0$ .

$- 4 {(x^{2} - 3 x + 2)}^{- 2} (2 x - 3)$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine $- 4$ and $\frac{1}{{(x^{2} - 3 x + 2)}^{2}}$ .

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

Step 5.1.2

Move the negative in front of the fraction.

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

Step 5.2

Reorder the factors of $- \frac{4}{{(x^{2} - 3 x + 2)}^{2}} (2 x - 3)$ .

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