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Differentiate (Tanx)÷(2cosx)​

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Answer:

[tex]\displaystyle \frac{d}{dx} = \frac{sin^2x + 1}{2cos^3x}[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Pre-Calculus

  • Trigonometric Functions

Calculus

Derivatives

Derivative Notation

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

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

Trig Derivative:                                                                                                       [tex]\displaystyle \frac{d}{dx}[tanu] = u'sec^2u[/tex]

Trig Derivative:                                                                                                      

[tex]\displaystyle \frac{d}{dx}[cosu] = -u'sinu[/tex]

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle y = \frac{tanx}{2cosx}[/tex]

Step 2: Differentiate

  1. [Derivative] Quotient Rule:                                                                               [tex]\displaystyle \frac{d}{dx} = \frac{\frac{d}{dx}[tanx](2cosx) - \frac{d}{dx}[2cosx](tanx)}{(2cosx)^2}[/tex]
  2. [Derivative] Simplify [Derivative Property - Multiplied Constant]:               [tex]\displaystyle \frac{d}{dx} = \frac{\frac{d}{dx}[tanx](2cosx) - 2\frac{d}{dx}[cosx](tanx)}{(2cosx)^2}[/tex]
  3. [Derivative] Evaluate [Trig Derivatives]:                                                        [tex]\displaystyle \frac{d}{dx} = \frac{sec^2x(2cosx) - (-2sinx)(tanx)}{(2cosx)^2}[/tex]
  4. [Derivative] Evaluate exponents:                                                                   [tex]\displaystyle \frac{d}{dx} = \frac{sec^2x(2cosx) - (-2sinx)(tanx)}{4cos^2x}[/tex]
  5. [Derivative] Multiply:                                                                                       [tex]\displaystyle \frac{d}{dx} = \frac{2secx + \frac{2sin^2x}{cosx}}{4cos^2x}[/tex]
  6. [Derivative] Add:                                                                                              [tex]\displaystyle \frac{d}{dx} = \frac{\frac{2sin^2x + 2}{cosx}}{4cos^2x}[/tex]
  7. [Derivative] Divide:                                                                                         [tex]\displaystyle \frac{d}{dx} = \frac{sin^2x + 1}{2cos^3x}[/tex]

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

Unit: Derivatives

Book: College Calculus 10e

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