what is the derivative of y=arccos(4x)/3x
To find the derivative of the function y = arccos(4x) / (3x), we can apply the quotient rule. The quotient rule states that if we have a function of the form f(x) / g(x), then its derivative can be found using the formula:
(f'(x) * g(x) - f(x) * g'(x)) / [g(x)]^2
Let's break down the steps involved in finding the derivative:
Step 1: Differentiate the numerator
The numerator of our function is arccos(4x). To differentiate arccos(4x), we need to remember two key facts:
- The derivative of arccos(x) is equal to -1 / sqrt(1 - x^2)
- The chain rule: if we have a composite function f(g(x)), then its derivative is given by f'(g(x)) * g'(x)
Using these facts, we find that the derivative of the numerator is:
-1 / sqrt(1 - (4x)^2) * (d/dx) (4x) = -1 / sqrt(1 - 16x^2) * 4
Step 2: Differentiate the denominator
The denominator of our function is 3x. To differentiate 3x, we simply need to apply the power rule, which states that the derivative of x^n is n * x^(n-1). In this case, the derivative of 3x is simply 3.
Step 3: Apply the quotient rule
Now that we have the derivatives of both the numerator and denominator, we can apply the quotient rule formula:
(f'(x) * g(x) - f(x) * g'(x)) / [g(x)]^2
Plugging in our values, we obtain the expression:
((-1 / sqrt(1 - 16x^2) * 4) * (3x) - (arccos(4x)) * 3) / (3x)^2
Simplifying further would involve algebraic manipulation, but this expression represents the derivative of y = arccos(4x) / (3x).