Calculus Examples

$y = \sqrt[5]{\frac{5 x}{3 x - 2}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{5 x}{3 x - 2}}$ as ${(\frac{5 x}{3 x - 2})}^{\frac{1}{5}}$ .

$y = {(\frac{5 x}{3 x - 2})}^{\frac{1}{5}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(\frac{5 x}{3 x - 2})}^{\frac{1}{5}})$

Step 3

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

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = \frac{5 x}{3 x - 2}$ .

Tap for more steps...

Step 4.1.1

To apply the Chain Rule, set $u$ as $\frac{5 x}{3 x - 2}$ .

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

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{5}$ .

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

Step 4.1.3

Replace all occurrences of $u$ with $\frac{5 x}{3 x - 2}$ .

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

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 4.3

Combine $- 1$ and $\frac{5}{5}$ .

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

Tap for more steps...

Step 4.5.1

Multiply $- 1$ by $5$ .

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

Step 4.5.2

Subtract $5$ from $1$ .

$\frac{1}{5} {(\frac{5 x}{3 x - 2})}^{\frac{- 4}{5}} \frac{d}{d x} [\frac{5 x}{3 x - 2}]$

Step 4.6

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{5} {(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{5 x}{3 x - 2}]$

Step 4.6.2

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

$\frac{1}{5} {(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} (5 \frac{d}{d x} [\frac{x}{3 x - 2}])$

Step 4.6.3

Simplify terms.

Tap for more steps...

Step 4.6.3.1

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} {(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{x}{3 x - 2}]$

Step 4.6.3.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 4.6.3.2.1

Cancel the common factor.

$\frac{5}{5} {(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{x}{3 x - 2}]$

Step 4.6.3.2.2

Rewrite the expression.

$1 {(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{x}{3 x - 2}]$

Step 4.6.3.3

Multiply ${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}}$ by $1$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{x}{3 x - 2}]$

Step 4.7

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

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{(3 x - 2) \frac{d}{d x} [x] - x \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.8

Differentiate.

Tap for more steps...

Step 4.8.1

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

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{(3 x - 2) \cdot 1 - x \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.8.2

Multiply $3 x - 2$ by $1$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.8.3

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

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.8.4

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

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.8.5

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

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x (3 \cdot 1 + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.8.6

Multiply $3$ by $1$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x (3 + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.8.7

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

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x (3 + 0)}{{(3 x - 2)}^{2}}$

Step 4.8.8

Simplify by adding terms.

Tap for more steps...

Step 4.8.8.1

Add $3$ and $0$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - x \cdot 3}{{(3 x - 2)}^{2}}$

Step 4.8.8.2

Multiply $3$ by $- 1$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{3 x - 2 - 3 x}{{(3 x - 2)}^{2}}$

Step 4.8.8.3

Subtract $3 x$ from $3 x$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{0 - 2}{{(3 x - 2)}^{2}}$

Step 4.8.8.4

Simplify the expression.

Tap for more steps...

Step 4.8.8.4.1

Subtract $2$ from $0$ .

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} \frac{- 2}{{(3 x - 2)}^{2}}$

Step 4.8.8.4.2

Move the negative in front of the fraction.

${(\frac{5 x}{3 x - 2})}^{- \frac{4}{5}} (- \frac{2}{{(3 x - 2)}^{2}})$

Step 4.9

Simplify.

Tap for more steps...

Step 4.9.1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{3 x - 2}{5 x})}^{\frac{4}{5}} (- \frac{2}{{(3 x - 2)}^{2}})$

Step 4.9.2

Apply the product rule to $\frac{3 x - 2}{5 x}$ .

$\frac{{(3 x - 2)}^{\frac{4}{5}}}{{(5 x)}^{\frac{4}{5}}} (- \frac{2}{{(3 x - 2)}^{2}})$

Step 4.9.3

Apply the product rule to $5 x$ .

$\frac{{(3 x - 2)}^{\frac{4}{5}}}{5^{\frac{4}{5}} x^{\frac{4}{5}}} (- \frac{2}{{(3 x - 2)}^{2}})$

Step 4.9.4

Combine terms.

Tap for more steps...

Step 4.9.4.1

Multiply $\frac{{(3 x - 2)}^{\frac{4}{5}}}{5^{\frac{4}{5}} x^{\frac{4}{5}}}$ by $\frac{2}{{(3 x - 2)}^{2}}$ .

$- \frac{{(3 x - 2)}^{\frac{4}{5}} \cdot 2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{2}}$

Step 4.9.4.2

Move $2$ to the left of ${(3 x - 2)}^{\frac{4}{5}}$ .

$- \frac{2 \cdot {(3 x - 2)}^{\frac{4}{5}}}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{2}}$

Step 4.9.4.3

Move ${(3 x - 2)}^{\frac{4}{5}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{2} {(3 x - 2)}^{- \frac{4}{5}}}$

Step 4.9.4.4

Multiply ${(3 x - 2)}^{2}$ by ${(3 x - 2)}^{- \frac{4}{5}}$ by adding the exponents.

Tap for more steps...

Step 4.9.4.4.1

Move ${(3 x - 2)}^{- \frac{4}{5}}$ .

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} ({(3 x - 2)}^{- \frac{4}{5}} {(3 x - 2)}^{2})}$

Step 4.9.4.4.2

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

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{- \frac{4}{5} + 2}}$

Step 4.9.4.4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{- \frac{4}{5} + 2 \cdot \frac{5}{5}}}$

Step 4.9.4.4.4

Combine $2$ and $\frac{5}{5}$ .

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{- \frac{4}{5} + \frac{2 \cdot 5}{5}}}$

Step 4.9.4.4.5

Combine the numerators over the common denominator.

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{\frac{- 4 + 2 \cdot 5}{5}}}$

Step 4.9.4.4.6

Simplify the numerator.

Tap for more steps...

Step 4.9.4.4.6.1

Multiply $2$ by $5$ .

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{\frac{- 4 + 10}{5}}}$

Step 4.9.4.4.6.2

Add $- 4$ and $10$ .

$- \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{\frac{6}{5}}}$

Step 5

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

$y' = - \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{\frac{6}{5}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{2}{5^{\frac{4}{5}} x^{\frac{4}{5}} {(3 x - 2)}^{\frac{6}{5}}}$