Question

Differentiate (Tanx)÷(2cosx)​

Answer 1

`\displaystyle (d)/(dx) = (sin^2x + 1)/(2cos^3x)`

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]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

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

Trig Derivative:
`\displaystyle (d)/(dx)[tanu] = u'sec^2u`

Trig Derivative:

`\displaystyle (d)/(dx)[cosu] = -u'sinu`

Explanation:

Step 1: Define

`\displaystyle y = (tanx)/(2cosx)`

Step 2: Differentiate

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

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

Unit: Derivatives

Book: College Calculus 10e