Calculus Examples

$f (x) = \frac{5}{3 x - 2}$

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 Constant Multiple Rule.

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Step 1.1.1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{3 x - 2}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{3 x - 2}]$ .

$5 \frac{d}{d x} [\frac{1}{3 x - 2}]$

Step 1.1.1.2

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

$5 \frac{d}{d x} [{(3 x - 2)}^{- 1}]$

Step 1.1.2

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 $g (x) = 3 x - 2$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $3 x - 2$ .

$5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [3 x - 2])$

Step 1.1.2.2

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

$5 (- u^{- 2} \frac{d}{d x} [3 x - 2])$

Step 1.1.2.3

Replace all occurrences of $u$ with $3 x - 2$ .

$5 (- {(3 x - 2)}^{- 2} \frac{d}{d x} [3 x - 2])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Multiply $- 1$ by $5$ .

$- 5 ({(3 x - 2)}^{- 2} \frac{d}{d x} [3 x - 2])$

Step 1.1.3.2

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

$- 5 {(3 x - 2)}^{- 2} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2])$

Step 1.1.3.3

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

$- 5 {(3 x - 2)}^{- 2} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])$

Step 1.1.3.4

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

$- 5 {(3 x - 2)}^{- 2} (3 \cdot 1 + \frac{d}{d x} [- 2])$

Step 1.1.3.5

Multiply $3$ by $1$ .

$- 5 {(3 x - 2)}^{- 2} (3 + \frac{d}{d x} [- 2])$

Step 1.1.3.6

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

$- 5 {(3 x - 2)}^{- 2} (3 + 0)$

Step 1.1.3.7

Simplify the expression.

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Step 1.1.3.7.1

Add $3$ and $0$ .

$- 5 {(3 x - 2)}^{- 2} \cdot 3$

Step 1.1.3.7.2

Multiply $3$ by $- 5$ .

$- 15 {(3 x - 2)}^{- 2}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$- 15 \frac{1}{{(3 x - 2)}^{2}}$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 15$ and $\frac{1}{{(3 x - 2)}^{2}}$ .

$\frac{- 15}{{(3 x - 2)}^{2}}$

Step 1.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{15}{{(3 x - 2)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{15}{{(3 x - 2)}^{2}}$ .

$- \frac{15}{{(3 x - 2)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{15}{{(3 x - 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{15}{{(3 x - 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$15 = 0$

Step 2.3

Since $15 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{15}{{(3 x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(3 x - 2)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Step 3.2.2

Solve for $x$ .

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Step 3.2.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 3.2.2.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 3.2.2.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 3.2.2.2.2

Simplify the left side.

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Step 3.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 4

Evaluate $\frac{5}{3 x - 2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{2}{3}$ .

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Step 4.1.1

Substitute $\frac{2}{3}$ for $x$ .

$\frac{5}{3 (\frac{2}{3}) - 2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{5}{3 (\frac{2}{3}) - 2}$

Step 4.1.2.1.2

Rewrite the expression.

$\frac{5}{2 - 2}$

Step 4.1.2.2

Subtract $2$ from $2$ .

$\frac{5}{0}$

Step 4.1.2.3

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

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found