Question

Find the derivative of cscx/3sinx

Answer 1

`\displaystyle (dy)/(dx) = (-\csc x(\cot x \sin x + \cos x))/(3\sin^2 x)`

General Formulas and Concepts:

Calculus

Differentiation

• 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))`

Explanation:

Step 1: Define

Identify

`\displaystyle y = (\csc x)/(3\sin x)`

Step 2: Differentiate

1. Derivative Property [Multiplied Constant]:
   `\displaystyle y' = (1)/(3) (d)/(dx)[(\csc x)/(\sin x)]`
2. Derivative Rule [Quotient Rule]:
   `\displaystyle y' = (1)/(3) \bigg( ((\csc x)' \sin x - \csc x (\sin x)')/(\sin^2 x) \bigg)`
3. Trigonometric Differentiation:
   `\displaystyle y' = (1)/(3) \bigg( (-\csc x \cot x \sin x - \csc x \cos x)/(\sin^2 x) \bigg)`
4. Factor:
   `\displaystyle y' = (1)/(3) \bigg( (-\csc x (\cot x \sin x + \cos x))/(\sin^2 x) \bigg)`
5. Simplify:
   `\displaystyle y' = (-\csc x (\cot x \sin x + \cos x))/(3\sin^2 x)`

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

Unit: Differentiation