Calculus Examples

${(1 - x)}^{- 2}$

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 $f (x) = x^{- 2}$ and $g (x) = 1 - x$ .

Tap for more steps...

Step 1.1.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 2 {(1 - x)}^{- 3} (0 + \frac{d}{d x} [- x])$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 1$ by $- 2$ .

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

Step 1.2.6

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

$2 {(1 - x)}^{- 3} \cdot 1$

Step 1.2.7

Multiply $2$ by $1$ .

$2 {(1 - x)}^{- 3}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

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

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

Step 2.1.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1.2.1

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

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

Step 2.1.2.2

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

Tap for more steps...

Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Multiply $3$ by $- 1$ .

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

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 $f (x) = x^{- 3}$ and $g (x) = 1 - x$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

Multiply $- 3$ by $2$ .

$- 6 ({(1 - x)}^{- 4} \frac{d}{d x} [1 - x])$

Step 2.3.2

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

$- 6 {(1 - x)}^{- 4} (\frac{d}{d x} [1] + \frac{d}{d x} [- x])$

Step 2.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 6 {(1 - x)}^{- 4} (0 + \frac{d}{d x} [- x])$

Step 2.3.4

Add $0$ and $\frac{d}{d x} [- x]$ .

$- 6 {(1 - x)}^{- 4} \frac{d}{d x} [- x]$

Step 2.3.5

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

$- 6 {(1 - x)}^{- 4} (- \frac{d}{d x} [x])$

Step 2.3.6

Multiply $- 1$ by $- 6$ .

$6 {(1 - x)}^{- 4} \frac{d}{d x} [x]$

Step 2.3.7

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

$6 {(1 - x)}^{- 4} \cdot 1$

Step 2.3.8

Multiply $6$ by $1$ .

$6 {(1 - x)}^{- 4}$

Step 2.4

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

$6 \frac{1}{{(1 - x)}^{4}}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Combine $6$ and $\frac{1}{{(1 - x)}^{4}}$ .

$\frac{6}{{(1 - x)}^{4}}$

Step 2.5.2

Reorder terms.

$f'' (x) = \frac{6}{{(- x + 1)}^{4}}$