Understanding the Derivative of arcsin(x 1) and Related Concepts

When dealing with derivatives in calculus, understanding the behavior of functions such as the arcsine (arcsin) function is fundamental. This article will delve into the derivative of arcsin(x 1), utilizing various techniques and explanations to ensure comprehensive understanding.

The Derivative of arcsin(x)

Before we tackle the derivative of arcsin(x 1), it's essential to understand the derivative of arcsin(x). There are multiple ways to derive this, but one common method involves using the inverse function theorem. The theorem states that if y f-1(x), then dy/dx 1 / [df(y)/dy]. Since arcsin(x) f-1(x) where f(x) sin(x), we can use the derivative of the sine function, dy/dx cos(y), to derive:

[ frac{d}{dx} arcsin(x) frac{1}{sqrt{1 - x^2}} ]

This can also be derived using the logarithmic identity and Euler's formula. One approach is to express arcsin(x) as (-i ln(i x sqrt{1-x^2})). By differentiating this expression using the chain rule and properties of the logarithm, we arrive at the same result:

[ frac{d}{dx} left[ -i ln(i x sqrt{1-x^2}) right] frac{1}{sqrt{1-x^2}} ]

Thus, the derivative of arcsin(x) is (frac{1}{sqrt{1-x^2}}).

Derivative of arcsin(x 1)

To find the derivative of (arcsin(x 1)), we use the chain rule. The chain rule states that if we have a composite function (h(x) f(g(x))), then the derivative is (h'(x) f'(g(x)) cdot g'(x)). Here, (f(x) arcsin(x)) and (g(x) x 1).

First, we define the functions: (f(x) arcsin(x)) with derivative (f'(x) frac{1}{sqrt{1-x^2}}) (g(x) x 1) with derivative (g'(x) 1)

Using the chain rule:

[ frac{d}{dx} arcsin(x 1) frac{1}{sqrt{1 - (x 1)^2}} cdot 1 ]

[ frac{1}{sqrt{1 - x^2 - 2x - 1}} frac{1}{sqrt{-x^2 - 2x}} ]

This simplifies to:

[ frac{1}{sqrt{-x^2 - 2x}} ]

Conclusion

Understanding the derivative of (arcsin(x 1)) involves applying the chain rule and leveraging the derivative of the arcsin function. This article has provided detailed steps and explanations to aid in the comprehension of these concepts, ensuring a clear and thorough understanding for students and professionals alike.

Further Reading

Differentiation Techniques in Calculus Inverse Trigonometric Functions Applications of the Chain Rule

By exploring these additional resources, you can gain a deeper insight into the various techniques and applications of derivatives, including those involving the arcsine function.