If f(x)=2cos(4ln(x)) , find f'(x)
f'(x) = -2sin(4lnx)(4/lnx)
To find the derivative of f(x) = 2cos(4ln(x)), we can use the chain rule.
Step 1: Rewrite the function using the chain rule notation:
f(x) = 2cos(u), where u = 4ln(x)
Step 2: Find the derivative of u with respect to x:
du/dx = (4/x)
Step 3: Apply the chain rule by multiplying the derivative of the outer function (cos(u)) by the derivative of the inner function (du/dx):
f'(x) = -2sin(u) * du/dx
Step 4: Substitute the values back in:
f'(x) = -2sin(4ln(x)) * (4/x)
So, the derivative of f(x) = 2cos(4ln(x)) is f'(x) = -2sin(4ln(x)) * (4/x).
To find the derivative of the function f(x) = 2cos(4ln(x)), we can use the chain rule.
The chain rule states that if we have a composition of functions, such as f(g(x)), the derivative is given by the product of the derivative of the outer function (f'(g(x))) and the derivative of the inner function (g'(x)).
In this case, the outer function is cosine (cos) and the inner function is 4ln(x).
To find f'(x), we need to find the derivative of f(x) with respect to x.
Let's break it down step by step.
Step 1: Find the derivative of the outer function (cos) with respect to its argument (4ln(x)).
The derivative of cos(u) with respect to u is -sin(u).
So, f'(x) = -sin(4ln(x)).
Step 2: Find the derivative of the inner function (4ln(x)) with respect to x.
The derivative of ln(x) with respect to x is 1/x.
Using the chain rule, the derivative of 4ln(x) with respect to x is 4(1/x) = 4/x.
Step 3: Multiply the results from Step 1 and Step 2.
f'(x) = -sin(4ln(x)) * (4/x).
So, the derivative of f(x) = 2cos(4ln(x)) with respect to x is f'(x) = -sin(4ln(x)) * (4/x).