Calculus Examples

$\frac{d^{2}}{d x^{2}} (\frac{10}{3 x^{3}})$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

$\frac{10}{3} \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

$\frac{10}{3} \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 1.2.2.2

Multiply $3$ by $- 1$ .

$\frac{10}{3} \frac{d}{d x} [x^{- 3}]$

Step 1.3

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

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

Step 1.4

Simplify terms.

Tap for more steps...

Step 1.4.1

Combine $- 3$ and $\frac{10}{3}$ .

$\frac{- 3 \cdot 10}{3} x^{- 4}$

Step 1.4.2

Multiply $- 3$ by $10$ .

$\frac{- 30}{3} x^{- 4}$

Step 1.4.3

Combine $\frac{- 30}{3}$ and $x^{- 4}$ .

$\frac{- 30 x^{- 4}}{3}$

Step 1.4.4

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

$\frac{- 30}{3 x^{4}}$

Step 1.4.5

Cancel the common factor of $- 30$ and $3$ .

Tap for more steps...

Step 1.4.5.1

Factor $3$ out of $- 30$ .

$\frac{3 \cdot - 10}{3 x^{4}}$

Step 1.4.5.2

Cancel the common factors.

Tap for more steps...

Step 1.4.5.2.1

Factor $3$ out of $3 x^{4}$ .

$\frac{3 \cdot - 10}{3 (x^{4})}$

Step 1.4.5.2.2

Cancel the common factor.

$\frac{3 \cdot - 10}{3 x^{4}}$

Step 1.4.5.2.3

Rewrite the expression.

$\frac{- 10}{x^{4}}$

Step 1.4.6

Move the negative in front of the fraction.

$f' (x) = - \frac{10}{x^{4}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10}{x^{4}}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 10 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

$- 10 \frac{d}{d x} [x^{4 \cdot - 1}]$

Step 2.2.2.2

Multiply $4$ by $- 1$ .

$- 10 \frac{d}{d x} [x^{- 4}]$

Step 2.3

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

$- 10 (- 4 x^{- 5})$

Step 2.4

Multiply $- 4$ by $- 10$ .

$40 x^{- 5}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

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

$40 \frac{1}{x^{5}}$

Step 2.5.2

Combine $40$ and $\frac{1}{x^{5}}$ .

$f'' (x) = \frac{40}{x^{5}}$