Question

Найти значение производной функции f(x)=x sin2x в точке x0=p

Answer 1

Finding the Derivative of \( f(x) = x + \sin(2x) \) at \( x_0 = p \)

To find the derivative of the function \( f(x) = x + \sin(2x) \) at the point \( x_0 = p \), we can use the rules of differentiation. The derivative of a function at a specific point gives the slope of the function at that point.

The derivative of \( f(x) = x + \sin(2x) \) can be calculated as follows:

\[ f'(x) = (x + \sin(2x))' = x' + (\sin(2x))' = 1 + 2\cos(2x) \]

Now, to find the value of the derivative at \( x_0 = p \), we substitute \( p \) into the derivative function:

\[ f'(p) = 1 + 2\cos(2p) \]

Therefore, the value of the derivative of \( f(x) = x + \sin(2x) \) at \( x_0 = p \) is \( 1 + 2\cos(2p) \).

Answer: The value of the derivative of the function \( f(x) = x + \sin(2x) \) at the point \( x_0 = p \) is \( 1 + 2\cos(2p) \).

I hope this helps! If you have any further questions, feel free to ask.