Solved Examples on Derivative of Arcsin x
Example 1: Find the derivative of y = arcsin (3x).
Solution:
Let f(x) = arcsin (3x).
We know that d/dx (arcsin x) = 1/√1 – x².
By chain rule,
d/dx(arcsin(3x)) = 1/√(1 – (3x)² · d/dx (3x)
= 1/ √(1 -9x²) · (3)
= 3/√(1 -9x²)
Hence, the derivative of y = arcsin (3x) is 3/√(1 -9x²).
Example 2: Find the derivative of y = arcsin (1/2x).
Solution:
Let f(x) = arcsin (1/2x).
We know that d/dx (arcsin x) = 1/√1 – x².
By chain rule,
d/dx(arcsin(1/2x)) = 1/√(1 – (1/2x)² · d/dx (1/2x)
= 1/ √(1 -(1/4x²) )· (-1/2x2)
= 1/√(4x2 – 1)/4x2 · (-1/2x2)
= -1/x√4x2 – 1
Hence, the derivative of y = arcsin (1/x) is -1/x√4x2 – 1.
Example 3: Find the derivative of y = x arcsin x.
Solution:
We have y = x arcsin x.
d/dx(arcsin(1/x)) = x · d/dx (arcsin x) + arcsin x · d/dx (x)
= x [1/√1-x²] + arcsin x (1)
= x/√1-x² + arcsin x
Hence, the derivative of y = arcsin (1/x) is x/√1-x² + arcsin x
Derivative of Arcsin
Derivative of Arcsin x is d/dx(arcsin x) = 1/√1-x². It is denoted by d/dx(arcsin x) or d/dx(sin-1x). Derivative of Arcsin refers to the process of finding the rate of change in Arcsin x function with respect to the independent variable. Derivative of Arcsin x is also known as differentiation of Arcsin.
In this article, we will learn about the derivative of Arcsin and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule method.
Table of Content
- What is Derivative in Math?
- What is Derivative of Arcsin x?
- Proof of Derivative of Arcsin x
- Solved Examples on Derivative of Arcsin x