To find the derivative of tan(2x), we will use the chain rule. The chain rule states that if you have a composite function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function.
First, let’s identify our functions: the outer function is tan(u), where u = 2x. The derivative of tan(u) is sec²(u). Now we need to differentiate the inner function, u = 2x, which has a derivative of 2.
Applying the chain rule, we can express the derivative of tan(2x) as:
Derivative of tan(2x) = sec²(2x) * (d/dx of 2x)
This simplifies to:
f'(x) = sec²(2x) * 2
Thus, the derivative of tan(2x) is:
f'(x) = 2 sec²(2x)