Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 3$ by $1$ .

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

Simplify the expression.

Tap for more steps...

Step 1.2.9.1

Add $1$ and $0$ .

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

Step 1.2.9.2

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

Tap for more steps...

Step 1.3.2.1

Simplify each term.

Tap for more steps...

Step 1.3.2.1.1

Expand $(x + 1) (2 x - 3)$ using the FOIL Method.

Tap for more steps...

Step 1.3.2.1.1.1

Apply the distributive property.

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

Step 1.3.2.1.1.2

Apply the distributive property.

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

Step 1.3.2.1.1.3

Apply the distributive property.

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

Step 1.3.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 1.3.2.1.2.1

Simplify each term.

Tap for more steps...

Step 1.3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.3.2.1.2.1.2.1

Move $x$ .

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

Step 1.3.2.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.2.1.2.1.3

Move $- 3$ to the left of $x$ .

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

Step 1.3.2.1.2.1.4

Multiply $2 x$ by $1$ .

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

Step 1.3.2.1.2.1.5

Multiply $- 3$ by $1$ .

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

Step 1.3.2.1.2.2

Add $- 3 x$ and $2 x$ .

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

Step 1.3.2.1.3

Multiply $- 3$ by $- 1$ .

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

Step 1.3.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 1.3.2.3

Add $- x$ and $3 x$ .

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

Step 1.3.3

Factor $x^{2} + 2 x - 3$ using the AC method.

Tap for more steps...

Step 1.3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.3.3.2

Write the factored form using these integers.

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $2$ .

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

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x + 3$ .

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

Step 2.4

Differentiate.

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

$\frac{{(x + 1)}^{2} ((x - 1) (1 + 0) + (x + 3) \frac{d}{d x} [x - 1]) - (x - 1) (x + 3) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.4.4

Simplify the expression.

Tap for more steps...

Step 2.4.4.1

Add $1$ and $0$ .

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

Step 2.4.4.2

Multiply $x - 1$ by $1$ .

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

Step 2.4.5

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

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

Step 2.4.6

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

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

Step 2.4.7

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

$\frac{{(x + 1)}^{2} (x - 1 + (x + 3) (1 + 0)) - (x - 1) (x + 3) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.4.8

Simplify by adding terms.

Tap for more steps...

Step 2.4.8.1

Add $1$ and $0$ .

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

Step 2.4.8.2

Multiply $x + 3$ by $1$ .

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

Step 2.4.8.3

Add $x$ and $x$ .

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

Step 2.4.8.4

Add $- 1$ and $3$ .

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

Step 2.5

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 $g (x) = x + 1$ .

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.6

Simplify with factoring out.

Tap for more steps...

Step 2.6.1

Multiply $2$ by $- 1$ .

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

Step 2.6.2

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

Tap for more steps...

Step 2.6.2.1

Factor $x + 1$ out of ${(x + 1)}^{2} (2 x + 2)$ .

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

Step 2.6.2.2

Factor $x + 1$ out of $- 2 (x - 1) (x + 3) ((x + 1) \frac{d}{d x} [x + 1])$ .

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

Step 2.6.2.3

Factor $x + 1$ out of $(x + 1) ((x + 1) (2 x + 2)) + (x + 1) (- 2 (x - 1) (x + 3) (\frac{d}{d x} [x + 1]))$ .

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

Step 2.7

Cancel the common factors.

Tap for more steps...

Step 2.7.1

Factor $x + 1$ out of ${(x + 1)}^{4}$ .

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify the expression.

Tap for more steps...

Step 2.11.1

Add $1$ and $0$ .

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

Step 2.11.2

Multiply $- 2$ by $1$ .

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

Step 2.12

Simplify.

Tap for more steps...

Step 2.12.1

Apply the distributive property.

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

Step 2.12.2

Simplify the numerator.

Tap for more steps...

Step 2.12.2.1

Simplify each term.

Tap for more steps...

Step 2.12.2.1.1

Expand $(x + 1) (2 x + 2)$ using the FOIL Method.

Tap for more steps...

Step 2.12.2.1.1.1

Apply the distributive property.

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

Step 2.12.2.1.1.2

Apply the distributive property.

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

Step 2.12.2.1.1.3

Apply the distributive property.

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

Step 2.12.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 2.12.2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.12.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.12.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.12.2.1.2.1.2.1

Move $x$ .

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

Step 2.12.2.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.12.2.1.2.1.3

Move $2$ to the left of $x$ .

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

Step 2.12.2.1.2.1.4

Multiply $2 x$ by $1$ .

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

Step 2.12.2.1.2.1.5

Multiply $2$ by $1$ .

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

Step 2.12.2.1.2.2

Add $2 x$ and $2 x$ .

$\frac{2 x^{2} + 4 x + 2 + (- 2 x - 2 \cdot - 1) (x + 3)}{{(x + 1)}^{3}}$

Step 2.12.2.1.3

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{2} + 4 x + 2 + (- 2 x + 2) (x + 3)}{{(x + 1)}^{3}}$

Step 2.12.2.1.4

Expand $(- 2 x + 2) (x + 3)$ using the FOIL Method.

Tap for more steps...

Step 2.12.2.1.4.1

Apply the distributive property.

$\frac{2 x^{2} + 4 x + 2 - 2 x (x + 3) + 2 (x + 3)}{{(x + 1)}^{3}}$

Step 2.12.2.1.4.2

Apply the distributive property.

$\frac{2 x^{2} + 4 x + 2 - 2 x \cdot x - 2 x \cdot 3 + 2 (x + 3)}{{(x + 1)}^{3}}$

Step 2.12.2.1.4.3

Apply the distributive property.

$\frac{2 x^{2} + 4 x + 2 - 2 x \cdot x - 2 x \cdot 3 + 2 x + 2 \cdot 3}{{(x + 1)}^{3}}$

Step 2.12.2.1.5

Simplify and combine like terms.

Tap for more steps...

Step 2.12.2.1.5.1

Simplify each term.

Tap for more steps...

Step 2.12.2.1.5.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.12.2.1.5.1.1.1

Move $x$ .

$\frac{2 x^{2} + 4 x + 2 - 2 (x \cdot x) - 2 x \cdot 3 + 2 x + 2 \cdot 3}{{(x + 1)}^{3}}$

Step 2.12.2.1.5.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 4 x + 2 - 2 x^{2} - 2 x \cdot 3 + 2 x + 2 \cdot 3}{{(x + 1)}^{3}}$

Step 2.12.2.1.5.1.2

Multiply $3$ by $- 2$ .

$\frac{2 x^{2} + 4 x + 2 - 2 x^{2} - 6 x + 2 x + 2 \cdot 3}{{(x + 1)}^{3}}$

Step 2.12.2.1.5.1.3

Multiply $2$ by $3$ .

$\frac{2 x^{2} + 4 x + 2 - 2 x^{2} - 6 x + 2 x + 6}{{(x + 1)}^{3}}$

Step 2.12.2.1.5.2

Add $- 6 x$ and $2 x$ .

$\frac{2 x^{2} + 4 x + 2 - 2 x^{2} - 4 x + 6}{{(x + 1)}^{3}}$

Step 2.12.2.2

Combine the opposite terms in $2 x^{2} + 4 x + 2 - 2 x^{2} - 4 x + 6$ .

Tap for more steps...

Step 2.12.2.2.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{4 x + 2 + 0 - 4 x + 6}{{(x + 1)}^{3}}$

Step 2.12.2.2.2

Add $4 x + 2$ and $0$ .

$\frac{4 x + 2 - 4 x + 6}{{(x + 1)}^{3}}$

Step 2.12.2.2.3

Subtract $4 x$ from $4 x$ .

$\frac{0 + 2 + 6}{{(x + 1)}^{3}}$

Step 2.12.2.2.4

Add $0$ and $2$ .

$\frac{2 + 6}{{(x + 1)}^{3}}$

Step 2.12.2.3

Add $2$ and $6$ .

$f'' (x) = \frac{8}{{(x + 1)}^{3}}$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 3.1.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{{(x + 1)}^{3}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{{(x + 1)}^{3}}]$ .

$8 \frac{d}{d x} [\frac{1}{{(x + 1)}^{3}}]$

Step 3.1.2

Apply basic rules of exponents.

Tap for more steps...

Step 3.1.2.1

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

$8 \frac{d}{d x} [{({(x + 1)}^{3})}^{- 1}]$

Step 3.1.2.2

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

Tap for more steps...

Step 3.1.2.2.1

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

$8 \frac{d}{d x} [{(x + 1)}^{3 \cdot - 1}]$

Step 3.1.2.2.2

Multiply $3$ by $- 1$ .

$8 \frac{d}{d x} [{(x + 1)}^{- 3}]$

Step 3.2

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 $g (x) = x + 1$ .

Tap for more steps...

Step 3.2.1

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

$8 (\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [x + 1])$

Step 3.2.2

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

$8 (- 3 u^{- 4} \frac{d}{d x} [x + 1])$

Step 3.2.3

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

$8 (- 3 {(x + 1)}^{- 4} \frac{d}{d x} [x + 1])$

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

Multiply $- 3$ by $8$ .

$- 24 ({(x + 1)}^{- 4} \frac{d}{d x} [x + 1])$

Step 3.3.2

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

$- 24 {(x + 1)}^{- 4} (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 3.3.3

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

$- 24 {(x + 1)}^{- 4} (1 + \frac{d}{d x} [1])$

Step 3.3.4

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

$- 24 {(x + 1)}^{- 4} (1 + 0)$

Step 3.3.5

Simplify the expression.

Tap for more steps...

Step 3.3.5.1

Add $1$ and $0$ .

$- 24 {(x + 1)}^{- 4} \cdot 1$

Step 3.3.5.2

Multiply $- 24$ by $1$ .

$- 24 {(x + 1)}^{- 4}$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

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

$- 24 \frac{1}{{(x + 1)}^{4}}$

Step 3.4.2

Combine terms.

Tap for more steps...

Step 3.4.2.1

Combine $- 24$ and $\frac{1}{{(x + 1)}^{4}}$ .

$\frac{- 24}{{(x + 1)}^{4}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{24}{{(x + 1)}^{4}}$