Question

help with 1 b please. using ln.​

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} = \frac{1}{(x - 2)^2\sqrt{\frac{x}{2 - x}}}$

General Formulas and Concepts:

Pre-Algebra

• Equality Properties

Algebra I

• Terms/Coefficients
• Factoring
• Exponential Rule [Root Rewrite]:                                                                 $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$

Algebra II

• Natural logarithms ln and Euler's number e
• Logarithmic Property [Exponential]:                                                             $\displaystyle log(a^b) = b \cdot log(a)$

Calculus

Differentiation

• Derivatives
• Derivative Notation
• Implicit Differentiation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

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

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

*Note:

You can simply just use the Quotient and Chain Rule to find the derivative instead of using ln.

Step 1: Define

Identify

$\displaystyle y = \sqrt{\frac{x}{2 - x}}$

Step 2: Rewrite

1. [Function] Exponential Rule [Root Rewrite]:                                               $\displaystyle y = \bigg( \frac{x}{2 - x} \bigg)^\bigg{\frac{1}{2}}$
2. [Equality Property] ln both sides:                                                                 $\displaystyle lny = ln \bigg[ \bigg( \frac{x}{2 - x} \bigg)^\bigg{\frac{1}{2}} \bigg]$
3. Logarithmic Property [Exponential]:                                                             $\displaystyle lny = \frac{1}{2}ln \bigg( \frac{x}{2 - x} \bigg)$

Step 3: Differentiate

1. Implicit Differentiation:                                                                                 $\displaystyle \frac{dy}{dx}[lny] = \frac{dy}{dx} \bigg[ \frac{1}{2}ln \bigg( \frac{x}{2 - x} \bigg) \bigg]$
2. Logarithmic Differentiation [Derivative Rule - Chain Rule]:                       $\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{1}{2} \bigg( \frac{1}{\frac{x}{2 - x}} \bigg) \frac{dy}{dx} \bigg[ \frac{x}{2 - x} \bigg]$
3. Chain Rule [Basic Power Rule]:                                                                     $\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{1}{2} \bigg( \frac{1}{\frac{x}{2 - x}} \bigg) \bigg[ \frac{2}{(x - 2)^2} \bigg]$
4. Simplify:                                                                                                         $\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{-1}{x(x - 2)}$
5. Isolate  $\displaystyle \frac{dy}{dx}$:                                                                                                     $\displaystyle \frac{dy}{dx} = \frac{-y}{x(x - 2)}$
6. Substitute in y [Derivative]:                                                                           $\displaystyle \frac{dy}{dx} = \frac{-\sqrt{\frac{x}{2 - x}}}{x(x - 2)}$
7. Rationalize:                                                                                                     $\displaystyle \frac{dy}{dx} = \frac{-\frac{x}{2 - x}}{x(x - 2)\sqrt{\frac{x}{2 - x}}}$
8. Rewrite:                                                                                                         $\displaystyle \frac{dy}{dx} = \frac{-x}{x(x - 2)(2 - x)\sqrt{\frac{x}{2 - x}}}$
9. Factor:                                                                                                           $\displaystyle \frac{dy}{dx} = \frac{-x}{-x(x - 2)^2\sqrt{\frac{x}{2 - x}}}$
10. Simplify:                                                                                                         $\displaystyle \frac{dy}{dx} = \frac{1}{(x - 2)^2\sqrt{\frac{x}{2 - x}}}$

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

Unit: Differentiation

Book: College Calculus 10e