Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4 {(x + 1)}^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{1}{{(x + 1)}^{\frac{3}{2}}}]$ .

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

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

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

Step 1.2.2

Multiply the exponents in ${({(x + 1)}^{\frac{3}{2}})}^{- 1}$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 1.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2.2.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{3}{2}$ .

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $- 3$ .

$- \frac{1}{4} (- \frac{3}{2} {(x + 1)}^{\frac{- 5}{2}} \frac{d}{d x} [x + 1])$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{1}{4} (- \frac{3}{2} {(x + 1)}^{- \frac{5}{2}} \frac{d}{d x} [x + 1])$

Step 7.2

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

$- \frac{1}{4} (- \frac{{(x + 1)}^{- \frac{5}{2}} \cdot 3}{2} \frac{d}{d x} [x + 1])$

Step 7.3

Simplify the expression.

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Step 7.3.1

Move $3$ to the left of ${(x + 1)}^{- \frac{5}{2}}$ .

$- \frac{1}{4} (- \frac{3 \cdot {(x + 1)}^{- \frac{5}{2}}}{2} \frac{d}{d x} [x + 1])$

Step 7.3.2

Move ${(x + 1)}^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{4} (- \frac{3}{2 {(x + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x + 1])$

Step 7.3.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{4}) (\frac{3}{2 {(x + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x + 1])$

Step 7.3.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} (\frac{3}{2 {(x + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x + 1])$

Step 7.4

Multiply $\frac{3}{2 {(x + 1)}^{\frac{5}{2}}}$ by $\frac{1}{4}$ .

$\frac{3}{2 {(x + 1)}^{\frac{5}{2}} \cdot 4} \frac{d}{d x} [x + 1]$

Step 7.5

Multiply $4$ by $2$ .

$\frac{3}{8 {(x + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x + 1]$

Step 8

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

$\frac{3}{8 {(x + 1)}^{\frac{5}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 9

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

$\frac{3}{8 {(x + 1)}^{\frac{5}{2}}} (1 + \frac{d}{d x} [1])$

Step 10

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

$\frac{3}{8 {(x + 1)}^{\frac{5}{2}}} (1 + 0)$

Step 11

Simplify the expression.

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Step 11.1

Add $1$ and $0$ .

$\frac{3}{8 {(x + 1)}^{\frac{5}{2}}} \cdot 1$

Step 11.2

Multiply $\frac{3}{8 {(x + 1)}^{\frac{5}{2}}}$ by $1$ .

$\frac{3}{8 {(x + 1)}^{\frac{5}{2}}}$