find f'(x) if f(x)=x^2cos^-1(ln�ãx)
To find the derivative of the function f(x) = x^2 * cos^(-1)(ln(x)), we can use the chain rule and differentiate each part separately.
Let's break down the function into its components:
1. The first component is x^2, which represents a power of x. The derivative of x^n (where n is a constant) with respect to x is given by d/dx(x^n) = n * x^(n-1).
So, the derivative of x^2 is d/dx(x^2) = 2x.
2. The second component is cos^(-1)(ln(x)). To differentiate it, we need to use the chain rule.
The derivative of cos^(-1)(u) with respect to u is -1 / (sqrt(1 - u^2)), where u = ln(x).
The derivative of ln(x) with respect to x is 1/x.
So, using the chain rule, the derivative of cos^(-1)(ln(x)) with respect to x is:
d/dx(cos^(-1)(ln(x))) = (-1 / sqrt(1 - (ln(x))^2)) * (1/x) = (-1 / (x * sqrt(1 - (ln(x))^2)).
Now, we multiply the derivatives of each component together to find the derivative of the entire function f(x):
f'(x) = (2x) * (-1 / (x * sqrt(1 - (ln(x))^2)))
Simplifying further, we have:
f'(x) = -2 / (sqrt(1 - (ln(x))^2))
Therefore, the derivative of f(x) = x^2 * cos^(-1)(ln(x)) is f'(x) = -2 / (sqrt(1 - (ln(x))^2)).