Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $1$ .

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Combine the numerators over the common denominator.

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

Step 1.3.6

Simplify the numerator.

Tap for more steps...

Step 1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.6.2

Subtract $3$ from $2$ .

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

Step 1.3.7

Move the negative in front of the fraction.

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

Step 1.3.8

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

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

Step 1.3.9

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

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

Step 1.3.10

Multiply $2$ by $- 3$ .

$f' (x) = 2 + \frac{- 6 x^{- \frac{1}{3}}}{3}$

Step 1.3.11

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

$f' (x) = 2 + \frac{- 6}{3 x^{\frac{1}{3}}}$

Step 1.3.12

Factor $3$ out of $- 6$ .

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

Step 1.3.13

Cancel the common factors.

Tap for more steps...

Step 1.3.13.1

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

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

Step 1.3.13.2

Cancel the common factor.

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

Step 1.3.13.3

Rewrite the expression.

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

Step 1.3.14

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (- \frac{2}{x^{\frac{1}{3}}})$

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

$f'' (x) = 0 - 2 \frac{d}{d x} \frac{1}{x^{\frac{1}{3}}}$

Step 2.2.2

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

$f'' (x) = 0 - 2 \frac{d}{d x} {(x^{\frac{1}{3}})}^{- 1}$

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 $x^{\frac{1}{3}}$ .

$f'' (x) = 0 - 2 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{3}}))$

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 $n = - 1$ .

$f'' (x) = 0 - 2 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{3}})$

Step 2.2.3.3

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

$f'' (x) = 0 - 2 (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} x^{\frac{1}{3}})$

Step 2.2.4

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

$f'' (x) = 0 - 2 (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

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, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = 0 - 2 (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.5.2

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

$f'' (x) = 0 - 2 (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.5.3

Move the negative in front of the fraction.

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.6

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

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}))$

Step 2.2.7

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

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 2.2.8

Combine the numerators over the common denominator.

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}))$

Step 2.2.9

Simplify the numerator.

Tap for more steps...

Step 2.2.9.1

Multiply $- 1$ by $3$ .

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}}))$

Step 2.2.9.2

Subtract $3$ from $1$ .

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}}))$

Step 2.2.10

Move the negative in front of the fraction.

$f'' (x) = 0 - 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}}))$

Step 2.2.11

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

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

Step 2.2.12

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

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

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.

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

Step 2.2.13.2

Combine the numerators over the common denominator.

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

Step 2.2.13.3

Subtract $2$ from $- 2$ .

$f'' (x) = 0 - 2 (- \frac{x^{\frac{- 4}{3}}}{3})$

Step 2.2.13.4

Move the negative in front of the fraction.

$f'' (x) = 0 - 2 (- \frac{x^{- \frac{4}{3}}}{3})$

Step 2.2.14

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

$f'' (x) = 0 - 2 (- \frac{1}{3 x^{\frac{4}{3}}})$

Step 2.2.15

Multiply $- 1$ by $- 2$ .

$f'' (x) = 0 + 2 (\frac{1}{3 x^{\frac{4}{3}}})$

Step 2.2.16

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

$f'' (x) = 0 + \frac{2}{3 x^{\frac{4}{3}}}$

Step 2.3

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

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

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3 x^{\frac{4}{3}}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$ .

$f''' (x) = \frac{2}{3} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{4}{3}}})$

Step 3.2

Apply basic rules of exponents.

Tap for more steps...

Step 3.2.1

Rewrite $\frac{1}{x^{\frac{4}{3}}}$ as ${(x^{\frac{4}{3}})}^{- 1}$ .

$f''' (x) = \frac{2}{3} \cdot \frac{d}{d x} ({(x^{\frac{4}{3}})}^{- 1})$

Step 3.2.2

Multiply the exponents in ${(x^{\frac{4}{3}})}^{- 1}$ .

Tap for more steps...

Step 3.2.2.1

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

$f''' (x) = \frac{2}{3} \cdot \frac{d}{d x} (x^{\frac{4}{3} \cdot - 1})$

Step 3.2.2.2

Multiply $\frac{4}{3} \cdot - 1$ .

Tap for more steps...

Step 3.2.2.2.1

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

$f''' (x) = \frac{2}{3} \cdot \frac{d}{d x} (x^{\frac{4 \cdot - 1}{3}})$

Step 3.2.2.2.2

Multiply $4$ by $- 1$ .

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

Step 3.2.2.3

Move the negative in front of the fraction.

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{4}{3}$ .

$f''' (x) = \frac{2}{3} \cdot (- \frac{4}{3} x^{- \frac{4}{3} - 1})$

Step 3.4

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

$f''' (x) = \frac{2}{3} \cdot (- \frac{4}{3} x^{- \frac{4}{3} - 1 \cdot \frac{3}{3}})$

Step 3.5

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

$f''' (x) = \frac{2}{3} \cdot (- \frac{4}{3} x^{- \frac{4}{3} + \frac{- 1 \cdot 3}{3}})$

Step 3.6

Combine the numerators over the common denominator.

$f''' (x) = \frac{2}{3} \cdot (- \frac{4}{3} x^{\frac{- 4 - 1 \cdot 3}{3}})$

Step 3.7

Simplify the numerator.

Tap for more steps...

Step 3.7.1

Multiply $- 1$ by $3$ .

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

Step 3.7.2

Subtract $3$ from $- 4$ .

$f''' (x) = \frac{2}{3} \cdot (- \frac{4}{3} x^{\frac{- 7}{3}})$

Step 3.8

Move the negative in front of the fraction.

$f''' (x) = \frac{2}{3} \cdot (- \frac{4}{3} x^{- \frac{7}{3}})$

Step 3.9

Combine $x^{- \frac{7}{3}}$ and $\frac{4}{3}$ .

$f''' (x) = \frac{2}{3} \cdot (- \frac{x^{- \frac{7}{3}} \cdot 4}{3})$

Step 3.10

Multiply $\frac{2}{3}$ by $\frac{x^{- \frac{7}{3}} \cdot 4}{3}$ .

$f''' (x) = - \frac{2 (x^{- \frac{7}{3}} \cdot 4)}{3 \cdot 3}$

Step 3.11

Multiply.

Tap for more steps...

Step 3.11.1

Multiply $4$ by $2$ .

$f''' (x) = - \frac{8 x^{- \frac{7}{3}}}{3 \cdot 3}$

Step 3.11.2

Multiply $3$ by $3$ .

$f''' (x) = - \frac{8 x^{- \frac{7}{3}}}{9}$

Step 3.11.3

Move $x^{- \frac{7}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f''' (x) = - \frac{8}{9 x^{\frac{7}{3}}}$

Step 4

Find the fourth derivative.

Tap for more steps...

Step 4.1

Since $- \frac{8}{9}$ is constant with respect to $x$ , the derivative of $- \frac{8}{9 x^{\frac{7}{3}}}$ with respect to $x$ is $- \frac{8}{9} \frac{d}{d x} [\frac{1}{x^{\frac{7}{3}}}]$ .

$f^{4} (x) = - \frac{8}{9} \cdot \frac{d}{d x} \frac{1}{x^{\frac{7}{3}}}$

Step 4.2

Apply basic rules of exponents.

Tap for more steps...

Step 4.2.1

Rewrite $\frac{1}{x^{\frac{7}{3}}}$ as ${(x^{\frac{7}{3}})}^{- 1}$ .

$f^{4} (x) = - \frac{8}{9} \cdot \frac{d}{d x} {(x^{\frac{7}{3}})}^{- 1}$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{7}{3}})}^{- 1}$ .

Tap for more steps...

Step 4.2.2.1

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

$f^{4} (x) = - \frac{8}{9} \cdot \frac{d}{d x} x^{\frac{7}{3} \cdot - 1}$

Step 4.2.2.2

Multiply $\frac{7}{3} \cdot - 1$ .

Tap for more steps...

Step 4.2.2.2.1

Combine $\frac{7}{3}$ and $- 1$ .

$f^{4} (x) = - \frac{8}{9} \cdot \frac{d}{d x} x^{\frac{7 \cdot - 1}{3}}$

Step 4.2.2.2.2

Multiply $7$ by $- 1$ .

$f^{4} (x) = - \frac{8}{9} \cdot \frac{d}{d x} x^{\frac{- 7}{3}}$

Step 4.2.2.3

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{8}{9} \cdot \frac{d}{d x} x^{- \frac{7}{3}}$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{7}{3}$ .

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{- \frac{7}{3} - 1})$

Step 4.4

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

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{- \frac{7}{3} - 1 \cdot \frac{3}{3}})$

Step 4.5

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

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{- \frac{7}{3} + \frac{- 1 \cdot 3}{3}})$

Step 4.6

Combine the numerators over the common denominator.

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{\frac{- 7 - 1 \cdot 3}{3}})$

Step 4.7

Simplify the numerator.

Tap for more steps...

Step 4.7.1

Multiply $- 1$ by $3$ .

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{\frac{- 7 - 3}{3}})$

Step 4.7.2

Subtract $3$ from $- 7$ .

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{\frac{- 10}{3}})$

Step 4.8

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{7}{3} x^{- \frac{10}{3}})$

Step 4.9

Combine $x^{- \frac{10}{3}}$ and $\frac{7}{3}$ .

$f^{4} (x) = - \frac{8}{9} \cdot (- \frac{x^{- \frac{10}{3}} \cdot 7}{3})$

Step 4.10

Multiply.

Tap for more steps...

Step 4.10.1

Multiply $- 1$ by $- 1$ .

$f^{4} (x) = 1 (\frac{8}{9}) \cdot \frac{x^{- \frac{10}{3}} \cdot 7}{3}$

Step 4.10.2

Multiply $\frac{8}{9}$ by $1$ .

$f^{4} (x) = \frac{8}{9} \cdot \frac{x^{- \frac{10}{3}} \cdot 7}{3}$

Step 4.11

Multiply $\frac{8}{9}$ by $\frac{x^{- \frac{10}{3}} \cdot 7}{3}$ .

$f^{4} (x) = \frac{8 (x^{- \frac{10}{3}} \cdot 7)}{9 \cdot 3}$

Step 4.12

Multiply.

Tap for more steps...

Step 4.12.1

Multiply $7$ by $8$ .

$f^{4} (x) = \frac{56 x^{- \frac{10}{3}}}{9 \cdot 3}$

Step 4.12.2

Multiply $9$ by $3$ .

$f^{4} (x) = \frac{56 x^{- \frac{10}{3}}}{27}$

Step 4.12.3

Move $x^{- \frac{10}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = \frac{56}{27 x^{\frac{10}{3}}}$