Finding the Derivative of cos^(1)((4x^(3))/(64)(3)/(4)x)
This article will walk you through the steps involved in finding the derivative of the inverse cosine function with a complex argument:
cos^(1)((4x^(3))/(64)(3)/(4)x)
We will use the chain rule and the derivative of the inverse cosine function to achieve this.
Understanding the Chain Rule
The chain rule is a fundamental concept in calculus that helps us differentiate composite functions. It states that the derivative of a composite function is equal to the product of the derivative of the outer function and the derivative of the inner function.
In mathematical terms:
If y = f(u) and u = g(x), then the derivative of y with respect to x is:
dy/dx = (dy/du) * (du/dx)
Finding the Derivative

Identify the inner and outer functions:
 Our outer function is f(u) = cos^(1)(u)
 Our inner function is u = g(x) = (4x^(3))/(64)(3)/(4)x

Find the derivatives of the inner and outer functions:
 d(f(u))/du = 1/√(1  u^2) (The derivative of the inverse cosine function)
 d(g(x))/dx = (12x^2)/64  3/4 = (3x^2)/16  3/4 (Using the power rule)

Apply the Chain Rule:
 d(cos^(1)((4x^(3))/(64)(3)/(4)x))/dx = 1/√(1  ((4x^(3))/(64)(3)/(4)x)^2) * ((3x^2)/16  3/4)

Simplify the expression:
 d(cos^(1)((4x^(3))/(64)(3)/(4)x))/dx = ((3x^2)/16  3/4) / √(1  ( (4x^3)/64  (3/4)x )^2)
Conclusion
The derivative of cos^(1)((4x^(3))/(64)(3)/(4)x) is:
((3x^2)/16  3/4) / √(1  ( (4x^3)/64  (3/4)x )^2)
This result can be further simplified or manipulated depending on the context of the problem.