What is the derivative of tan(2x)?
Understand the Problem
The question is asking for the derivative of the function tan(2x). To solve this, we will apply the chain rule of differentiation.
Answer
2 \sec^2(2x)
Answer for screen readers
The final answer is $2 \sec^2(2x)$
Steps to Solve

Identify the outer and inner functions for the chain rule
The function tan(2x) can be broken down into the outer function $f(u) = an(u)$ and the inner function $u = 2x$

Differentiate the outer function
The derivative of the outer function $f(u) = an(u)$ is $f'(u) = ext{sec}^2(u)$

Differentiate the inner function
The derivative of the inner function $u = 2x$ is $u' = 2$

Apply the chain rule
According to the chain rule, the derivative of $f(g(x))$ is $f'(g(x)) imes g'(x)$.
Here, it becomes:
$$\frac{d}{dx}[ an(2x)] = \sec^2(2x) \times 2$$

Simplify the result
Therefore, the derivative of $ an(2x)$ is:
$$\frac{d}{dx}[ an(2x)] = 2 \sec^2(2x)$$
The final answer is $2 \sec^2(2x)$
More Information
The secant function (sec) is the reciprocal of the cosine function. The chain rule helps by breaking down the differentiation of composite functions into manageable parts.
Tips
A common mistake is forgetting to apply the chain rule completely, i.e., not multiplying by the derivative of the inner function.