Calculus Examples

$y = {(5 x + 30)}^{\frac{2}{3}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 5 x + 30$ .

Tap for more steps...

Step 1.1.1

To apply the Chain Rule, set $u$ as $5 x + 30$ .

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [5 x + 30]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [5 x + 30]$

Step 1.1.3

Replace all occurrences of $u$ with $5 x + 30$ .

$\frac{2}{3} {(5 x + 30)}^{\frac{2}{3} - 1} \frac{d}{d x} [5 x + 30]$

Step 1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} {(5 x + 30)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [5 x + 30]$

Step 1.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{2}{3} {(5 x + 30)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [5 x + 30]$

Step 1.4

Combine the numerators over the common denominator.

$\frac{2}{3} {(5 x + 30)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [5 x + 30]$

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} {(5 x + 30)}^{\frac{2 - 3}{3}} \frac{d}{d x} [5 x + 30]$

Step 1.5.2

Subtract $3$ from $2$ .

$\frac{2}{3} {(5 x + 30)}^{\frac{- 1}{3}} \frac{d}{d x} [5 x + 30]$

Step 1.6

Combine fractions.

Tap for more steps...

Step 1.6.1

Move the negative in front of the fraction.

$\frac{2}{3} {(5 x + 30)}^{- \frac{1}{3}} \frac{d}{d x} [5 x + 30]$

Step 1.6.2

Combine $\frac{2}{3}$ and ${(5 x + 30)}^{- \frac{1}{3}}$ .

$\frac{2 {(5 x + 30)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [5 x + 30]$

Step 1.6.3

Move ${(5 x + 30)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} \frac{d}{d x} [5 x + 30]$

Step 1.7

By the Sum Rule, the derivative of $5 x + 30$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [30]$ .

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} (\frac{d}{d x} [5 x] + \frac{d}{d x} [30])$

Step 1.8

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

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} (5 \frac{d}{d x} [x] + \frac{d}{d x} [30])$

Step 1.9

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

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} (5 \cdot 1 + \frac{d}{d x} [30])$

Step 1.10

Multiply $5$ by $1$ .

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} (5 + \frac{d}{d x} [30])$

Step 1.11

Since $30$ is constant with respect to $x$ , the derivative of $30$ with respect to $x$ is $0$ .

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} (5 + 0)$

Step 1.12

Combine fractions.

Tap for more steps...

Step 1.12.1

Add $5$ and $0$ .

$\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}} \cdot 5$

Step 1.12.2

Combine $\frac{2}{3 {(5 x + 30)}^{\frac{1}{3}}}$ and $5$ .

$\frac{2 \cdot 5}{3 {(5 x + 30)}^{\frac{1}{3}}}$

Step 1.12.3

Multiply $2$ by $5$ .

$f' (x) = \frac{10}{3 {(5 x + 30)}^{\frac{1}{3}}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 2.1.1

Since $\frac{10}{3}$ is constant with respect to $x$ , the derivative of $\frac{10}{3 {(5 x + 30)}^{\frac{1}{3}}}$ with respect to $x$ is $\frac{10}{3} \frac{d}{d x} [\frac{1}{{(5 x + 30)}^{\frac{1}{3}}}]$ .

$\frac{10}{3} \frac{d}{d x} [\frac{1}{{(5 x + 30)}^{\frac{1}{3}}}]$

Step 2.1.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1.2.1

Rewrite $\frac{1}{{(5 x + 30)}^{\frac{1}{3}}}$ as ${({(5 x + 30)}^{\frac{1}{3}})}^{- 1}$ .

$\frac{10}{3} \frac{d}{d x} [{({(5 x + 30)}^{\frac{1}{3}})}^{- 1}]$

Step 2.1.2.2

Multiply the exponents in ${({(5 x + 30)}^{\frac{1}{3}})}^{- 1}$ .

Tap for more steps...

Step 2.1.2.2.1

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

$\frac{10}{3} \frac{d}{d x} [{(5 x + 30)}^{\frac{1}{3} \cdot - 1}]$

Step 2.1.2.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{10}{3} \frac{d}{d x} [{(5 x + 30)}^{\frac{- 1}{3}}]$

Step 2.1.2.2.3

Move the negative in front of the fraction.

$\frac{10}{3} \frac{d}{d x} [{(5 x + 30)}^{- \frac{1}{3}}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{3}}$ and $g (x) = 5 x + 30$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $5 x + 30$ .

$\frac{10}{3} (\frac{d}{d u} [u^{- \frac{1}{3}}] \frac{d}{d x} [5 x + 30])$

Step 2.2.2

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

$\frac{10}{3} (- \frac{1}{3} u^{- \frac{1}{3} - 1} \frac{d}{d x} [5 x + 30])$

Step 2.2.3

Replace all occurrences of $u$ with $5 x + 30$ .

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{- \frac{1}{3} - 1} \frac{d}{d x} [5 x + 30])$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [5 x + 30])$

Step 2.4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [5 x + 30])$

Step 2.5

Combine the numerators over the common denominator.

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{\frac{- 1 - 1 \cdot 3}{3}} \frac{d}{d x} [5 x + 30])$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $3$ .

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{\frac{- 1 - 3}{3}} \frac{d}{d x} [5 x + 30])$

Step 2.6.2

Subtract $3$ from $- 1$ .

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{\frac{- 4}{3}} \frac{d}{d x} [5 x + 30])$

Step 2.7

Combine fractions.

Tap for more steps...

Step 2.7.1

Move the negative in front of the fraction.

$\frac{10}{3} (- \frac{1}{3} {(5 x + 30)}^{- \frac{4}{3}} \frac{d}{d x} [5 x + 30])$

Step 2.7.2

Combine ${(5 x + 30)}^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$\frac{10}{3} (- \frac{{(5 x + 30)}^{- \frac{4}{3}}}{3} \frac{d}{d x} [5 x + 30])$

Step 2.7.3

Move ${(5 x + 30)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10}{3} (- \frac{1}{3 {(5 x + 30)}^{\frac{4}{3}}} \frac{d}{d x} [5 x + 30])$

Step 2.7.4

Multiply $\frac{10}{3}$ by $\frac{1}{3 {(5 x + 30)}^{\frac{4}{3}}}$ .

$- \frac{10}{3 (3 {(5 x + 30)}^{\frac{4}{3}})} \frac{d}{d x} [5 x + 30]$

Step 2.7.5

Multiply $3$ by $3$ .

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} \frac{d}{d x} [5 x + 30]$

Step 2.8

By the Sum Rule, the derivative of $5 x + 30$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [30]$ .

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} (\frac{d}{d x} [5 x] + \frac{d}{d x} [30])$

Step 2.9

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

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} (5 \frac{d}{d x} [x] + \frac{d}{d x} [30])$

Step 2.10

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

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} (5 \cdot 1 + \frac{d}{d x} [30])$

Step 2.11

Multiply $5$ by $1$ .

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} (5 + \frac{d}{d x} [30])$

Step 2.12

Since $30$ is constant with respect to $x$ , the derivative of $30$ with respect to $x$ is $0$ .

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} (5 + 0)$

Step 2.13

Combine fractions.

Tap for more steps...

Step 2.13.1

Add $5$ and $0$ .

$- \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}} \cdot 5$

Step 2.13.2

Multiply $5$ by $- 1$ .

$- 5 \frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}}$

Step 2.13.3

Combine $- 5$ and $\frac{10}{9 {(5 x + 30)}^{\frac{4}{3}}}$ .

$\frac{- 5 \cdot 10}{9 {(5 x + 30)}^{\frac{4}{3}}}$

Step 2.13.4

Simplify the expression.

Tap for more steps...

Step 2.13.4.1

Multiply $- 5$ by $10$ .

$\frac{- 50}{9 {(5 x + 30)}^{\frac{4}{3}}}$

Step 2.13.4.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{50}{9 {(5 x + 30)}^{\frac{4}{3}}}$