Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [3 x^{\frac{2}{3}}]$ .

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify the numerator.

Tap for more steps...

Step 1.2.6.1

Multiply $- 1$ by $3$ .

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

Step 1.2.6.2

Subtract $3$ from $2$ .

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Multiply $2$ by $3$ .

$\frac{6 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- 2 x]$

Step 1.2.11

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

$\frac{6}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 2 x]$

Step 1.2.12

Factor $3$ out of $6$ .

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

Step 1.2.13

Cancel the common factors.

Tap for more steps...

Step 1.2.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

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

Step 1.2.13.2

Cancel the common factor.

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

Step 1.2.13.3

Rewrite the expression.

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

Step 1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 2$ by $1$ .

$f' (x) = \frac{2}{x^{\frac{1}{3}}} - 2$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $\frac{2}{x^{\frac{1}{3}}} - 2$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{x^{\frac{1}{3}}}] + \frac{d}{d x} [- 2]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [\frac{2}{x^{\frac{1}{3}}}]$ .

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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 $g (x) = x^{\frac{1}{3}}$ .

Tap for more steps...

Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{3}}$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

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

Step 2.2.4

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

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

Step 2.2.5

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 2}$ .

Tap for more steps...

Step 2.2.5.1

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

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

Step 2.2.5.2

Combine $\frac{1}{3}$ and $- 2$ .

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

Step 2.2.5.3

Move the negative in front of the fraction.

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Combine the numerators over the common denominator.

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

Step 2.2.9

Simplify the numerator.

Tap for more steps...

Step 2.2.9.1

Multiply $- 1$ by $3$ .

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

Step 2.2.9.2

Subtract $3$ from $1$ .

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

Step 2.2.10

Move the negative in front of the fraction.

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

Step 2.2.11

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

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

Step 2.2.12

Combine $\frac{x^{- \frac{2}{3}}}{3}$ and $x^{- \frac{2}{3}}$ .

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

Step 2.2.13

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

Tap for more steps...

Step 2.2.13.1

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

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

Step 2.2.13.2

Combine the numerators over the common denominator.

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

Step 2.2.13.3

Subtract $2$ from $- 2$ .

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

Step 2.2.13.4

Move the negative in front of the fraction.

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

Step 2.2.14

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

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

Step 2.2.15

Multiply $- 1$ by $2$ .

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

Step 2.2.16

Combine $- 2$ and $\frac{1}{3 x^{\frac{4}{3}}}$ .

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

Step 2.2.17

Move the negative in front of the fraction.

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

Step 2.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Add $- \frac{2}{3 x^{\frac{4}{3}}}$ and $0$ .

$f'' (x) = - \frac{2}{3 x^{\frac{4}{3}}}$