what is the derivative of y=arccos(4x)/3x
To find the derivative of the given function y = arccos(4x) / (3x), we can use the quotient rule. The quotient rule states that if you have a function in the form f(x) / g(x), where f(x) and g(x) are both differentiable, then the derivative of y with respect to x is given by:
dy/dx = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2
Let's apply this rule to our function step by step:
Step 1: Identify f(x) and g(x)
In our case, f(x) = arccos(4x) and g(x) = 3x.
Step 2: Find f'(x) and g'(x)
To find f'(x), we need the derivative of arccos(4x) with respect to x. The derivative of arccos(u) is -1 / sqrt(1 - u^2), so in this case, f'(x) = -1 / sqrt(1 - (4x)^2).
To find g'(x), we need the derivative of 3x, which is simply 3.
Step 3: Apply the derivative formula
Now we have all the necessary components to apply the quotient rule:
dy/dx = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2
= (3x * (-1 / sqrt(1 - (4x)^2)) - arccos(4x) * 3) / (3x)^2
Simplifying this expression, we get:
dy/dx = (-3 / sqrt(1 - (4x)^2) - 3arccos(4x)) / (9x^2)