Calculus Examples

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

Step 1

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 $g (x) = 4 x + 3$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x + 3$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $4 x + 3$ .

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

Step 2

By the Sum Rule, the derivative of $4 x + 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$ .

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

Step 3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $4$ by $1$ .

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

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

Step 4.2

Add $4$ and $0$ .

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

Step 5

Combine terms.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $- 1$ by $2$ .

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

Step 5.4.2

Subtract $2$ from $1$ .

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

Step 5.5

Move the negative in front of the fraction.

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Move $4$ to the left of ${(4 x + 3)}^{- \frac{1}{2}}$ .

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

Step 5.9

Move ${(4 x + 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.10

Factor $2$ out of $4$ .

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

Step 5.11

Cancel the common factors.

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Step 5.11.1

Factor $2$ out of $2 {(4 x + 3)}^{\frac{1}{2}}$ .

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

Step 5.11.2

Cancel the common factor.

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

Step 5.11.3

Rewrite the expression.

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