The derivative of the function sin(2x) can be found using the chain rule, which is a fundamental theorem in calculus used for differentiating composite functions.
To find the derivative, we first recognize that sin(2x) is a composition of two functions: the sine function and the linear function 2x.
1. **Differentiate the outer function**: The derivative of sin(u) with respect to u is cos(u), where in this case, u = 2x.
2. **Differentiate the inner function**: The derivative of 2x with respect to x is 2.
3. **Apply the chain rule**: Now, according to the chain rule, the derivative of sin(2x) with respect to x is:
cos(2x) * (2)
Putting it all together, we have:
f'(x) = 2 * cos(2x)
So, the derivative of sin(2x) is 2 * cos(2x).
This result gives you the rate of change of sin(2x) at any point x and is useful in various applications such as physics, engineering, and financial modeling.