derivative of f(x)=2 cot (4e^x)
To find the derivative of f(x) = 2cot(4e^x), we will use the chain rule.
First, let's find the derivative of the inner function, g(x) = 4e^x:
g'(x) = 4e^x
Now, let's find the derivative of f(x) using the chain rule:
f'(x) = -2csc^2(4e^x) * (4e^x)'
= -2csc^2(4e^x) * 4e^x
Therefore, the derivative of f(x) = 2cot(4e^x) is f'(x) = -8e^x csc^2(4e^x).
To find the derivative of f(x) = 2 cot (4e^x), we can use the chain rule.
Let's start by finding the derivative of the function inside the cotangent:
f'(x) = 2 * d/dx(cot (4e^x))
The derivative of cot (u) is given by -csc^2 (u) * du/dx. In this case, u = 4e^x.
So, d/dx (cot (4e^x)) = -csc^2 (4e^x) * d/dx (4e^x)
Now, let's find the derivative of 4e^x:
d/dx (4e^x) = 4 * d/dx (e^x)
The derivative of e^x is simply e^x, so:
d/dx (4e^x) = 4 * e^x
Let's substitute this back into our equation:
f'(x) = 2 * (-csc^2 (4e^x) * 4 * e^x)
Now simplify this further:
f'(x) = -8e^x * csc^2 (4e^x)
Therefore, the derivative of f(x) = 2 cot (4e^x) is:
f'(x) = -8e^x * csc^2 (4e^x)
To find the derivative of f(x) = 2 cot (4e^x), we will need to use the chain rule. The chain rule states that if we have a function g(x) inside another function f(x), the derivative of f(g(x)) with respect to x is f'(g(x)) times g'(x).
Let's begin by finding the derivative of the function g(x) = 4e^x. The derivative of e^x with respect to x is e^x itself. So, the derivative of g(x) can be written as g'(x) = 4e^x.
Next, we need to find the derivative of f(x) with respect to x, which involves the derivative of the cot function. The derivative of cot(x) is equal to -csc^2(x), where csc(x) represents the cosecant function.
Combining these derivatives, we have:
f'(x) = 2 * -csc^2 (4e^x) * g'(x)
Substituting the derivative of g(x) into the equation, we get:
f'(x) = 2 * -csc^2 (4e^x) * 4e^x
Now, simplified further, we have:
f'(x) = -8e^x * csc^2 (4e^x)
Therefore, the derivative of f(x) = 2 cot (4e^x) is f'(x) = -8e^x * csc^2 (4e^x).