Calculus Examples

$x + \sqrt{7 x - 3}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x + \sqrt{7 x - 3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{7 x - 3}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{7 x - 3}]$

Step 1.2

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

$1 + \frac{d}{d x} [\sqrt{7 x - 3}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt{7 x - 3}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x - 3}$ as ${(7 x - 3)}^{\frac{1}{2}}$ .

$1 + \frac{d}{d x} [{(7 x - 3)}^{\frac{1}{2}}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 7 x - 3$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $7 x - 3$ .

$1 + \frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [7 x - 3]$

Step 2.2.2

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

$1 + \frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [7 x - 3]$

Step 2.2.3

Replace all occurrences of $u$ with $7 x - 3$ .

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x - 3]$

Step 2.3

By the Sum Rule, the derivative of $7 x - 3$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 3]$ .

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} - 1} (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 3])$

Step 2.4

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} - 1} (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.5

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

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} - 1} (7 \cdot 1 + \frac{d}{d x} [- 3])$

Step 2.6

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

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} - 1} (7 \cdot 1 + 0)$

Step 2.7

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

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (7 \cdot 1 + 0)$

Step 2.8

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

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (7 \cdot 1 + 0)$

Step 2.9

Combine the numerators over the common denominator.

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1 - 1 \cdot 2}{2}} (7 \cdot 1 + 0)$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{1 - 2}{2}} (7 \cdot 1 + 0)$

Step 2.10.2

Subtract $2$ from $1$ .

$1 + \frac{1}{2} {(7 x - 3)}^{\frac{- 1}{2}} (7 \cdot 1 + 0)$

Step 2.11

Move the negative in front of the fraction.

$1 + \frac{1}{2} {(7 x - 3)}^{- \frac{1}{2}} (7 \cdot 1 + 0)$

Step 2.12

Multiply $7$ by $1$ .

$1 + \frac{1}{2} {(7 x - 3)}^{- \frac{1}{2}} (7 + 0)$

Step 2.13

Add $7$ and $0$ .

$1 + \frac{1}{2} {(7 x - 3)}^{- \frac{1}{2}} \cdot 7$

Step 2.14

Combine $\frac{1}{2}$ and ${(7 x - 3)}^{- \frac{1}{2}}$ .

$1 + \frac{{(7 x - 3)}^{- \frac{1}{2}}}{2} \cdot 7$

Step 2.15

Combine $\frac{{(7 x - 3)}^{- \frac{1}{2}}}{2}$ and $7$ .

$1 + \frac{{(7 x - 3)}^{- \frac{1}{2}} \cdot 7}{2}$

Step 2.16

Move $7$ to the left of ${(7 x - 3)}^{- \frac{1}{2}}$ .

$1 + \frac{7 \cdot {(7 x - 3)}^{- \frac{1}{2}}}{2}$

Step 2.17

Move ${(7 x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 + \frac{7}{2 {(7 x - 3)}^{\frac{1}{2}}}$