How can I find the derivative of the function y = tan(2x)?

To find the derivative of the function y = tan(2x), we will apply the chain rule, which is a fundamental technique in calculus for finding the derivative of composite functions.

Step 1: Understand the Components
In our case, we have a composition of two functions: the tangent function and the inner function 2x. The derivative of tan(u) where u = 2x can be differentiated using the chain rule.

Step 2: Apply the Chain Rule
The chain rule states that if you have a function f(g(x)), then the derivative f'(g(x)) * g'(x) can be used. For our function, we can denote:

• f(u) = tan(u) with the derivative f'(u) = sec²(u)
• g(x) = 2x with the derivative g'(x) = 2

Now, applying the chain rule:

y' = f'(g(x)) * g'(x) = sec2(2x) * 2

Step 3: Combine the Results
Thus, the derivative of the function can be written as:

y' = 2 * sec2(2x)

Step 4: Final Result
So, the final answer for the derivative of y = tan(2x) is:

y' = 2sec2(2x)

This means that wherever you have the function y = tan(2x), the slope of the tangent line at any point can be found using y’ = 2sec²(2x). Always remember to keep practicing your derivatives, as it helps solidify your understanding of how functions behave and change!