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

  1. 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 $u = 2x$

  2. Differentiate the outer function

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

  3. Differentiate the inner function

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

  4. 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:

    $$\frac{d}{dx}[ an(2x)] = \sec^2(2x) \times 2$$

  5. 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.

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