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(d)/(dx) Cos^(-1)((4x^(3))/(64)-(3)/(4)x) =

3 min read Jun 16, 2024
(d)/(dx) Cos^(-1)((4x^(3))/(64)-(3)/(4)x) =

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

  1. 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

  2. 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)

  3. 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)

  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.

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