Question

Find from first principle the derivative of 3x+5/√x​

Answer 1

Answer:

$\displaystyle \frac{d}{dx} = \frac{3x - 5}{2x^\bigg{\frac{3}{2}}}$

General Formulas and Concepts:

Algebra I

• Exponential Rule [Powering]:                                                                          $\displaystyle (b^m)^n = b^{m \cdot n}$
• Exponential Rule [Rewrite]:                                                                              $\displaystyle b^{-m} = \frac{1}{b^m}$
• Exponential Rule [Root Rewrite]:                                                                     $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$

Calculus

Derivatives

Derivative Notation

Derivative Property [Addition/Subtraction]:                                                            $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                               $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \frac{3x + 5}{\sqrt{x}}$

Step 2: Differentiate

1. Rewrite [Exponential Rule - Root Rewrite]:                                                     $\displaystyle \frac{3x + 5}{x^\bigg{\frac{1}{2}}}$
2. Quotient Rule:                                                                                                   $\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})\frac{d}{dx}[3x + 5] - \frac{d}{dx}[x^\bigg{\frac{1}{2}}](3x + 5)}{(x^\bigg{\frac{1}{2}})^2}$
3. Simplify [Exponential Rule - Powering]:                                                          $\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})\frac{d}{dx}[3x + 5] - \frac{d}{dx}[x^\bigg{\frac{1}{2}}](3x + 5)}{x}$
4. Basic Power Rule [Derivative Property - Addition/Subtraction]:                   $\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})(3x^{1 - 1} + 0) - (\frac{1}{2}x^\bigg{\frac{1}{2} - 1})(3x + 5)}{x}$
5. Simplify:                                                                                                             $\displaystyle \frac{d}{dx} = \frac{3x^\bigg{\frac{1}{2}} - (\frac{1}{2}x^\bigg{\frac{-1}{2}})(3x + 5)}{x}$
6. Rewrite [Exponential Rule - Rewrite]:                                                              $\displaystyle \frac{d}{dx} = \frac{3x^\bigg{\frac{1}{2}} - (\frac{1}{2x^{\frac{1}{2}}})(3x + 5)}{x}$
7. Rewrite [Exponential Rule - Root Rewrite]:                                                     $\displaystyle \frac{d}{dx} = \frac{3\sqrt{x} - (\frac{1}{2\sqrt{x}})(3x + 5)}{x}$
8. Simplify [Rationalize]:                                                                                       $\displaystyle \frac{d}{dx} = \frac{3x - 5}{2x^\bigg{\frac{3}{2}}}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Derivatives

Book: College Calculus 10e