What is the derivative of 2sin(2x)?
Understand the Problem
The question is asking for the derivative of the function 2sin(2x) with respect to x. To solve this, we will apply the chain rule of differentiation. The derivative of sin(u) is cos(u) multiplied by the derivative of u, where u = 2x in this case.
Answer
$4\cos(2x)$
Answer for screen readers
The derivative of the function $2\sin(2x)$ with respect to $x$ is $4\cos(2x)$.
Steps to Solve
- Identify the function and its components
The function we need to differentiate is $f(x) = 2 \sin(2x)$. We can see that it involves a sine function and has an inner function $u = 2x$.
- Apply the chain rule
Using the chain rule, we differentiate the outer function $\sin(u)$ and then multiply by the derivative of the inner function $u$. The chain rule states that if $f(u) = \sin(u)$, then $$ f'(u) = \cos(u) \cdot \frac{du}{dx} $$
- Differentiate the outer function
The derivative of the outer function is: $$ f'(x) = 2 \cdot \cos(2x) $$
- Differentiate the inner function
Now we find the derivative of the inner function $u = 2x$: $$ \frac{du}{dx} = 2 $$
- Combine the derivatives
Now we combine these results using the chain rule: $$ \frac{d}{dx}(2 \sin(2x)) = 2 \cdot \cos(2x) \cdot 2 = 4 \cos(2x) $$
The derivative of the function $2\sin(2x)$ with respect to $x$ is $4\cos(2x)$.
More Information
The chain rule is a fundamental concept in calculus, used to differentiate composite functions. Derivatives provide information about the rate of change of functions and have wide applications in physics, engineering, and economics.
Tips
- Forgetting to multiply by the derivative of the inner function, which can lead to an incorrect answer. Always remember to apply the chain rule fully.
- Neglecting to simplify the final expression could lead to confusion in the interpretation of results.
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