Calculus Examples

$f (x) = \frac{1}{x^{7}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Apply basic rules of exponents.

Tap for more steps...

Step 1.1.1

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

$\frac{d}{d x} [{(x^{7})}^{- 1}]$

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

$\frac{d}{d x} [x^{7 \cdot - 1}]$

Step 1.1.2.2

Multiply $7$ by $- 1$ .

$\frac{d}{d x} [x^{- 7}]$

Step 1.2

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

$- 7 x^{- 8}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

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

$- 7 \frac{1}{x^{8}}$

Step 1.3.2

Combine terms.

Tap for more steps...

Step 1.3.2.1

Combine $- 7$ and $\frac{1}{x^{8}}$ .

$\frac{- 7}{x^{8}}$

Step 1.3.2.2

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

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

$- 7 \frac{d}{d x} [\frac{1}{x^{8}}]$

Step 2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1

Rewrite $\frac{1}{x^{8}}$ as ${(x^{8})}^{- 1}$ .

$- 7 \frac{d}{d x} [{(x^{8})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(x^{8})}^{- 1}$ .

Tap for more steps...

Step 2.2.2.1

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

$- 7 \frac{d}{d x} [x^{8 \cdot - 1}]$

Step 2.2.2.2

Multiply $8$ by $- 1$ .

$- 7 \frac{d}{d x} [x^{- 8}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 8$ .

$- 7 (- 8 x^{- 9})$

Step 2.4

Multiply $- 8$ by $- 7$ .

$56 x^{- 9}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

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

$56 \frac{1}{x^{9}}$

Step 2.5.2

Combine $56$ and $\frac{1}{x^{9}}$ .

$f'' (x) = \frac{56}{x^{9}}$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

Since $56$ is constant with respect to $x$ , the derivative of $\frac{56}{x^{9}}$ with respect to $x$ is $56 \frac{d}{d x} [\frac{1}{x^{9}}]$ .

$56 \frac{d}{d x} [\frac{1}{x^{9}}]$

Step 3.2

Apply basic rules of exponents.

Tap for more steps...

Step 3.2.1

Rewrite $\frac{1}{x^{9}}$ as ${(x^{9})}^{- 1}$ .

$56 \frac{d}{d x} [{(x^{9})}^{- 1}]$

Step 3.2.2

Multiply the exponents in ${(x^{9})}^{- 1}$ .

Tap for more steps...

Step 3.2.2.1

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

$56 \frac{d}{d x} [x^{9 \cdot - 1}]$

Step 3.2.2.2

Multiply $9$ by $- 1$ .

$56 \frac{d}{d x} [x^{- 9}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 9$ .

$56 (- 9 x^{- 10})$

Step 3.4

Multiply $- 9$ by $56$ .

$- 504 x^{- 10}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

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

$- 504 \frac{1}{x^{10}}$

Step 3.5.2

Combine terms.

Tap for more steps...

Step 3.5.2.1

Combine $- 504$ and $\frac{1}{x^{10}}$ .

$\frac{- 504}{x^{10}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{504}{x^{10}}$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- \frac{504}{x^{10}}$ .

$- \frac{504}{x^{10}}$