Calculus Examples

$\frac{6 x}{{(x - 9)}^{2}}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{{(x - 9)}^{2}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{{(x - 9)}^{2}}]$ .

$6 \frac{d}{d x} [\frac{x}{{(x - 9)}^{2}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(x - 9)}^{2}$ .

$6 \frac{{(x - 9)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x - 9)}^{2}]}{{({(x - 9)}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(x - 9)}^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 \frac{{(x - 9)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x - 9)}^{2}]}{{(x - 9)}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$6 \frac{{(x - 9)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x - 9)}^{2}]}{{(x - 9)}^{4}}$

Step 3.2

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

$6 \frac{{(x - 9)}^{2} \cdot 1 - x \frac{d}{d x} [{(x - 9)}^{2}]}{{(x - 9)}^{4}}$

Step 3.3

Multiply ${(x - 9)}^{2}$ by $1$ .

$6 \frac{{(x - 9)}^{2} - x \frac{d}{d x} [{(x - 9)}^{2}]}{{(x - 9)}^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 9$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x - 9$ .

$6 \frac{{(x - 9)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 9])}{{(x - 9)}^{4}}$

Step 4.2

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

$6 \frac{{(x - 9)}^{2} - x (2 u \frac{d}{d x} [x - 9])}{{(x - 9)}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $x - 9$ .

$6 \frac{{(x - 9)}^{2} - x (2 (x - 9) \frac{d}{d x} [x - 9])}{{(x - 9)}^{4}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

$6 \frac{{(x - 9)}^{2} - 2 x ((x - 9) \frac{d}{d x} [x - 9])}{{(x - 9)}^{4}}$

Step 5.2

Factor $x - 9$ out of ${(x - 9)}^{2} - 2 x ((x - 9) \frac{d}{d x} [x - 9])$ .

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Step 5.2.1

Factor $x - 9$ out of ${(x - 9)}^{2}$ .

$6 \frac{(x - 9) (x - 9) - 2 x ((x - 9) \frac{d}{d x} [x - 9])}{{(x - 9)}^{4}}$

Step 5.2.2

Factor $x - 9$ out of $- 2 x ((x - 9) \frac{d}{d x} [x - 9])$ .

$6 \frac{(x - 9) (x - 9) + (x - 9) (- 2 x (\frac{d}{d x} [x - 9]))}{{(x - 9)}^{4}}$

Step 5.2.3

Factor $x - 9$ out of $(x - 9) (x - 9) + (x - 9) (- 2 x (\frac{d}{d x} [x - 9]))$ .

$6 \frac{(x - 9) (x - 9 - 2 x (\frac{d}{d x} [x - 9]))}{{(x - 9)}^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $x - 9$ out of ${(x - 9)}^{4}$ .

$6 \frac{(x - 9) (x - 9 - 2 x (\frac{d}{d x} [x - 9]))}{(x - 9) {(x - 9)}^{3}}$

Step 6.2

Cancel the common factor.

$6 \frac{(x - 9) (x - 9 - 2 x (\frac{d}{d x} [x - 9]))}{(x - 9) {(x - 9)}^{3}}$

Step 6.3

Rewrite the expression.

$6 \frac{x - 9 - 2 x (\frac{d}{d x} [x - 9])}{{(x - 9)}^{3}}$

Step 7

By the Sum Rule, the derivative of $x - 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 9]$ .

$6 \frac{x - 9 - 2 x (\frac{d}{d x} [x] + \frac{d}{d x} [- 9])}{{(x - 9)}^{3}}$

Step 8

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

$6 \frac{x - 9 - 2 x (1 + \frac{d}{d x} [- 9])}{{(x - 9)}^{3}}$

Step 9

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$6 \frac{x - 9 - 2 x (1 + 0)}{{(x - 9)}^{3}}$

Step 10

Simplify terms.

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Step 10.1

Add $1$ and $0$ .

$6 \frac{x - 9 - 2 x \cdot 1}{{(x - 9)}^{3}}$

Step 10.2

Multiply $- 2$ by $1$ .

$6 \frac{x - 9 - 2 x}{{(x - 9)}^{3}}$

Step 10.3

Subtract $2 x$ from $x$ .

$6 \frac{- x - 9}{{(x - 9)}^{3}}$

Step 10.4

Combine $6$ and $\frac{- x - 9}{{(x - 9)}^{3}}$ .

$\frac{6 (- x - 9)}{{(x - 9)}^{3}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{6 (- x) + 6 \cdot - 9}{{(x - 9)}^{3}}$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $- 1$ by $6$ .

$\frac{- 6 x + 6 \cdot - 9}{{(x - 9)}^{3}}$

Step 11.2.2

Multiply $6$ by $- 9$ .

$\frac{- 6 x - 54}{{(x - 9)}^{3}}$

Step 11.3

Factor $6$ out of $- 6 x - 54$ .

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Step 11.3.1

Factor $6$ out of $- 6 x$ .

$\frac{6 (- x) - 54}{{(x - 9)}^{3}}$

Step 11.3.2

Factor $6$ out of $- 54$ .

$\frac{6 (- x) + 6 (- 9)}{{(x - 9)}^{3}}$

Step 11.3.3

Factor $6$ out of $6 (- x) + 6 (- 9)$ .

$\frac{6 (- x - 9)}{{(x - 9)}^{3}}$

Step 11.4

Factor $- 1$ out of $- x$ .

$\frac{6 (- (x) - 9)}{{(x - 9)}^{3}}$

Step 11.5

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{6 (- (x) - 1 (9))}{{(x - 9)}^{3}}$

Step 11.6

Factor $- 1$ out of $- (x) - 1 (9)$ .

$\frac{6 (- (x + 9))}{{(x - 9)}^{3}}$

Step 11.7

Rewrite $- (x + 9)$ as $- 1 (x + 9)$ .

$\frac{6 (- 1 (x + 9))}{{(x - 9)}^{3}}$

Step 11.8

Move the negative in front of the fraction.

$- \frac{6 (x + 9)}{{(x - 9)}^{3}}$