Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $g (x) = {(x - 9)}^{2}$ .

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

Step 1.1.2

Differentiate using the Power Rule.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

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 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.4

Simplify with factoring out.

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Step 1.1.4.1

Multiply $2$ by $- 1$ .

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

Step 1.1.4.2

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

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Step 1.1.4.2.1

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

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

Step 1.1.4.2.2

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

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

Step 1.1.4.2.3

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

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

Step 1.1.5

Cancel the common factors.

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Step 1.1.5.1

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

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

Step 1.1.5.2

Cancel the common factor.

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

Step 1.1.5.3

Rewrite the expression.

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

Step 1.1.6

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

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

Simplify by adding terms.

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Step 1.1.9.1

Add $1$ and $0$ .

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

Step 1.1.9.2

Multiply $- 2$ by $1$ .

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

Step 1.1.9.3

Subtract $2 x$ from $x$ .

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

Step 1.1.10

Simplify.

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Step 1.1.10.1

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

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

Step 1.1.10.2

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

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

Step 1.1.10.3

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

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

Step 1.1.10.4

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

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

Step 1.1.10.5

Move the negative in front of the fraction.

$f' (x) = - \frac{x + 9}{{(x - 9)}^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x + 9}{{(x - 9)}^{3}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x + 9}{{(x - 9)}^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x + 9}{{(x - 9)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$x + 9 = 0$

Step 2.3

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x + 9}{{(x - 9)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x - 9)}^{3} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 3.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 4

Evaluate $\frac{x}{{(x - 9)}^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 9$ .

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Step 4.1.1

Substitute $- 9$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the denominator.

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Step 4.1.2.1.1

Subtract $9$ from $- 9$ .

$\frac{- 9}{{(- 18)}^{2}}$

Step 4.1.2.1.2

Raise $- 18$ to the power of $2$ .

$\frac{- 9}{324}$

Step 4.1.2.2

Reduce the expression by cancelling the common factors.

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Step 4.1.2.2.1

Cancel the common factor of $- 9$ and $324$ .

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Step 4.1.2.2.1.1

Factor $9$ out of $- 9$ .

$\frac{9 (- 1)}{324}$

Step 4.1.2.2.1.2

Cancel the common factors.

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Step 4.1.2.2.1.2.1

Factor $9$ out of $324$ .

$\frac{9 \cdot - 1}{9 \cdot 36}$

Step 4.1.2.2.1.2.2

Cancel the common factor.

$\frac{9 \cdot - 1}{9 \cdot 36}$

Step 4.1.2.2.1.2.3

Rewrite the expression.

$\frac{- 1}{36}$

Step 4.1.2.2.2

Move the negative in front of the fraction.

$- \frac{1}{36}$

Step 4.2

Evaluate at $x = 9$ .

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Step 4.2.1

Substitute $9$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $9$ from $9$ .

$\frac{9}{0^{2}}$

Step 4.2.2.2

Raising $0$ to any positive power yields $0$ .

$\frac{9}{0}$

Step 4.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(- 9, - \frac{1}{36})$

Step 5