Calculus Examples

$f (x) = \frac{3}{x - 5}$

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1.1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x - 5}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x - 5}]$ .

$f' (x) = 3 \frac{d}{d x} (\frac{1}{x - 5})$

Step 1.1.1.2

Rewrite $\frac{1}{x - 5}$ as ${(x - 5)}^{- 1}$ .

$f' (x) = 3 \frac{d}{d x} ({(x - 5)}^{- 1})$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x - 5$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$f' (x) = 3 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x - 5))$

Step 1.1.2.2

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

$f' (x) = 3 (- u^{- 2} \frac{d}{d x} (x - 5))$

Step 1.1.2.3

Replace all occurrences of $u$ with $x - 5$ .

$f' (x) = 3 (- {(x - 5)}^{- 2} \frac{d}{d x} (x - 5))$

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

Multiply $- 1$ by $3$ .

$f' (x) = - 3 ({(x - 5)}^{- 2} \frac{d}{d x} (x - 5))$

Step 1.1.3.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$f' (x) = - 3 {(x - 5)}^{- 2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 5))$

Step 1.1.3.3

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

$f' (x) = - 3 {(x - 5)}^{- 2} (1 + \frac{d}{d x} (- 5))$

Step 1.1.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f' (x) = - 3 {(x - 5)}^{- 2} (1 + 0)$

Step 1.1.3.5

Simplify the expression.

Tap for more steps...

Step 1.1.3.5.1

Add $1$ and $0$ .

$f' (x) = - 3 {(x - 5)}^{- 2} \cdot 1$

Step 1.1.3.5.2

Multiply $- 3$ by $1$ .

$f' (x) = - 3 {(x - 5)}^{- 2}$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

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

$f' (x) = - 3 \frac{1}{{(x - 5)}^{2}}$

Step 1.1.4.2

Combine terms.

Tap for more steps...

Step 1.1.4.2.1

Combine $- 3$ and $\frac{1}{{(x - 5)}^{2}}$ .

$f' (x) = \frac{- 3}{{(x - 5)}^{2}}$

Step 1.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{3}{{(x - 5)}^{2}}$

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.2.1.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{{(x - 5)}^{2}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{{(x - 5)}^{2}}]$ .

$f'' (x) = - 3 \frac{d}{d x} \frac{1}{{(x - 5)}^{2}}$

Step 1.2.1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1.2.1

Rewrite $\frac{1}{{(x - 5)}^{2}}$ as ${({(x - 5)}^{2})}^{- 1}$ .

$f'' (x) = - 3 \frac{d}{d x} {({(x - 5)}^{2})}^{- 1}$

Step 1.2.1.2.2

Multiply the exponents in ${({(x - 5)}^{2})}^{- 1}$ .

Tap for more steps...

Step 1.2.1.2.2.1

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

$f'' (x) = - 3 \frac{d}{d x} {(x - 5)}^{2 \cdot - 1}$

Step 1.2.1.2.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = - 3 \frac{d}{d x} {(x - 5)}^{- 2}$

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = x - 5$ .

Tap for more steps...

Step 1.2.2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$f'' (x) = - 3 (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (x - 5))$

Step 1.2.2.2

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

$f'' (x) = - 3 (- 2 u^{- 3} \frac{d}{d x} (x - 5))$

Step 1.2.2.3

Replace all occurrences of $u$ with $x - 5$ .

$f'' (x) = - 3 (- 2 {(x - 5)}^{- 3} \frac{d}{d x} (x - 5))$

Step 1.2.3

Differentiate.

Tap for more steps...

Step 1.2.3.1

Multiply $- 2$ by $- 3$ .

$f'' (x) = 6 ({(x - 5)}^{- 3} \frac{d}{d x} (x - 5))$

Step 1.2.3.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$f'' (x) = 6 {(x - 5)}^{- 3} (\frac{d}{d x} (x) + \frac{d}{d x} (- 5))$

Step 1.2.3.3

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

$f'' (x) = 6 {(x - 5)}^{- 3} (1 + \frac{d}{d x} (- 5))$

Step 1.2.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f'' (x) = 6 {(x - 5)}^{- 3} (1 + 0)$

Step 1.2.3.5

Simplify the expression.

Tap for more steps...

Step 1.2.3.5.1

Add $1$ and $0$ .

$f'' (x) = 6 {(x - 5)}^{- 3} \cdot 1$

Step 1.2.3.5.2

Multiply $6$ by $1$ .

$f'' (x) = 6 {(x - 5)}^{- 3}$

Step 1.2.4

Simplify.

Tap for more steps...

Step 1.2.4.1

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

$f'' (x) = 6 (\frac{1}{{(x - 5)}^{3}})$

Step 1.2.4.2

Combine $6$ and $\frac{1}{{(x - 5)}^{3}}$ .

$f'' (x) = \frac{6}{{(x - 5)}^{3}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{6}{{(x - 5)}^{3}}$ .

$\frac{6}{{(x - 5)}^{3}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{6}{{(x - 5)}^{3}} = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to $0$ .

$\frac{6}{{(x - 5)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$6 = 0$

Step 2.3

Since $6 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points