Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{3}{x - 2}$ .

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

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\frac{3}{x - 2}$ .

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

Step 3.2

Differentiate using the Constant Multiple Rule.

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Step 3.2.1

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

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $3$ by $3$ .

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

Step 3.2.2.2

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

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

Step 3.3

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

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $x - 2$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u_{2}$ with $x - 2$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

Multiply $- 1$ by $9$ .

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Simplify the expression.

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Step 3.4.5.1

Add $1$ and $0$ .

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

Step 3.4.5.2

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

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

Step 3.5

Simplify.

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Step 3.5.1

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

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

Step 3.5.2

Apply the product rule to $\frac{3}{x - 2}$ .

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

Step 3.5.3

Combine terms.

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Step 3.5.3.1

Raise $3$ to the power of $2$ .

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

Step 3.5.3.2

Combine $- 9$ and $\frac{9}{{(x - 2)}^{2}}$ .

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

Step 3.5.3.3

Multiply $- 9$ by $9$ .

$\frac{- 81}{{(x - 2)}^{2}} \cdot \frac{1}{{(x - 2)}^{2}}$

Step 3.5.3.4

Move the negative in front of the fraction.

$- \frac{81}{{(x - 2)}^{2}} \cdot \frac{1}{{(x - 2)}^{2}}$

Step 3.5.3.5

Multiply $\frac{1}{{(x - 2)}^{2}}$ by $\frac{81}{{(x - 2)}^{2}}$ .

$- \frac{81}{{(x - 2)}^{2} {(x - 2)}^{2}}$

Step 3.5.3.6

Multiply ${(x - 2)}^{2}$ by ${(x - 2)}^{2}$ by adding the exponents.

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Step 3.5.3.6.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{81}{{(x - 2)}^{2 + 2}}$

Step 3.5.3.6.2

Add $2$ and $2$ .

$- \frac{81}{{(x - 2)}^{4}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{81}{{(x - 2)}^{4}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{81}{{(x - 2)}^{4}}$