Question

Given f(x)=arctan(2x), find f’(x) in simplest form

Answer 1

The derivative of the function f(x) = arctan(2x) is found using the chain rule. The resulting derivative is f'(x) = 2/(1+4x^2).

Step-by-step explanation:

To find the derivative f'(x) of the function f(x) = arctan(2x), we apply the chain rule. The derivative of arctan(x), also known as the inverse tangent function, is 1/(1+x2). Now, let's implement the chain rule where the outer function is arctan(u), where u = 2x, and the inner function is u itself. Therefore, the derivative of u with respect to x is 2, as d(2x)/dx = 2.

Combining these results gives us the derivative of the composite function

f'(x) = d(arctan(2x))/dx = d(arctan(u))/du * du/dx = 1/(1+u2) * 2 = 2/(1+(2x)2) = 2/(1+4x2)

Answer 2

Step-by-step explanation:

`f(x)=tan^(-1) 2x\\let f(x)=y\\y=tan^(-1)2x\\tan y=2x\\diff. w.r.t.x\\sec^2y((dy)/(dx) )=2\\(dy)/(dx) =(2)/(sec^2 y) =(2)/(1+tan^2y) =(2)/(1+(2x)^2) \\f'(x)=(2)/(1+4x^2)`