Calculus Examples

$f (x) = \frac{3}{x + 2} - \sqrt{x - 3}$

Step 1

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

$\frac{d}{d x} [\frac{3}{x + 2}] + \frac{d}{d x} [- \sqrt{x - 3}]$

Step 2

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

Tap for more steps...

Step 2.1

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

$3 \frac{d}{d x} [\frac{1}{x + 2}] + \frac{d}{d x} [- \sqrt{x - 3}]$

Step 2.2

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

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

Step 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 $g (x) = x + 2$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $x + 2$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u_{1}$ with $x + 2$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $1$ and $0$ .

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

Step 2.8

Multiply $- 1$ by $1$ .

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

Step 2.9

Multiply $- 1$ by $3$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

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

Step 3.2

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

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 3$ .

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $x - 3$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u_{2}$ with $x - 3$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.8

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

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

Step 3.9

Combine the numerators over the common denominator.

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

Step 3.10

Simplify the numerator.

Tap for more steps...

Step 3.10.1

Multiply $- 1$ by $2$ .

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

Step 3.10.2

Subtract $2$ from $1$ .

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

Step 3.11

Move the negative in front of the fraction.

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

Step 3.12

Add $1$ and $0$ .

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Move ${(x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Combine terms.

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

Move the negative in front of the fraction.

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