Mixed examples : Differentiating inverse trig functions

Mixed examples : Differentiating inverse trig functions

Differentiating Inverse Trig Functions

Basics

• The inverse trigonometric functions are the arcsine, arccosine, and arctangent, denoted as asin, acos, atan respectively.
• In calculus, differentiation refers to the process of finding the derivative of a function.

Expressions of Derivatives

• The derivative of asin(x) is expressed as 1 / sqrt(1 - x²).
• The derivative of acos(x) is expressed as -1 / sqrt(1 - x²).
• The derivative of atan(x) is expressed as 1 / (1 + x²).

Applying Chain Rule

• When differentiating composite functions of the form asin(f(x)), acos(f(x)), atan(f(x)), use the chain rule.
• The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Example Problems

Example 1: Differentiating asin(x²)

• The function is asin(x²).
• Using chain rule, the derivative will be the derivative of the outer function times the derivative of the inner function.
• Therefore, the derivative is 2x / sqrt(1 - x⁴).

Example 2: Differentiating acos(5x)

• The function is acos(5x).
• Using chain rule, the derivative is the derivative of the outer function times the derivative of the inner function.
• Therefore, the derivative is -5 / sqrt(1 - 25x²).

Example 3: Differentiating atan(3x²)

• The function is atan(3x²).
• Using chain rule, the derivative is the derivative of the outer function times the derivative of the inner function.
• Therefore, the derivative is 6x / (1 + 9x⁴).

General Tips

• Pay attention to signs and squares in your calculations.
• Practice a variety of problems to solidify your understanding.
• Remember to always keep the chain rule in mind as it is fundamental to solving these problems.
• Practice makes perfect, the more problems you solve, the more proficient you will become.