Calculus Examples

$v = \frac{2 x}{{(x + 1)}^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} v' = \frac{d}{d x} (\frac{2 x}{{(x + 1)}^{2}})$

Step 2

The derivative of $v'$ with respect to $x$ is $v''$ .

$v''$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Multiply $2$ by $2$ .

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

Step 3.3.2

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

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

Step 3.3.3

Multiply ${(x + 1)}^{2}$ by $1$ .

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

Step 3.4

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 $g (x) = x + 1$ .

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $2$ by $- 1$ .

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

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

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

Step 3.5.2.3

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

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

Step 3.6

Cancel the common factors.

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Step 3.6.1

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

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

Step 3.6.2

Cancel the common factor.

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

Step 3.6.3

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Simplify terms.

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Step 3.10.1

Add $1$ and $0$ .

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

Step 3.10.2

Multiply $- 2$ by $1$ .

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

Step 3.10.3

Subtract $2 x$ from $x$ .

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

Step 3.10.4

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

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Multiply $- 1$ by $2$ .

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

Step 3.11.2.2

Multiply $2$ by $1$ .

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

Step 3.11.3

Factor $2$ out of $- 2 x + 2$ .

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Step 3.11.3.1

Factor $2$ out of $- 2 x$ .

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

Step 3.11.3.2

Factor $2$ out of $2$ .

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

Step 3.11.3.3

Factor $2$ out of $2 (- x) + 2 (1)$ .

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

Step 3.11.4

Factor $- 1$ out of $- x$ .

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

Step 3.11.5

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.11.6

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

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

Step 3.11.7

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

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

Step 3.11.8

Move the negative in front of the fraction.

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $v'$ with $\frac{d v}{d x}$ .

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