To find the first two derivatives of the function y = 2 sin(x) cos(x), we can start by using a trigonometric identity to simplify our work. We know from the double angle identity that:
sin(2x) = 2 sin(x) cos(x)
This means we can rewrite our function as:
y = sin(2x)
Now that we have simplified our function, we can easily find the derivatives:
First Derivative:
Using the chain rule, the first derivative of y = sin(2x) is:
dy/dx = 2 cos(2x)
This step involves taking the derivative of sin(u) where u = 2x, which gives us cos(u) * du/dx, and since du/dx = 2, we multiply it by 2.
Second Derivative:
Now, to find the second derivative, we differentiate the first derivative:
d²y/dx² = d/dx (2 cos(2x))
Again applying the chain rule, we get:
d²y/dx² = 2 * (-sin(2x)) * 2 = -4 sin(2x)
In summary, the first two derivatives of 2 sin(x) cos(x) are:
- First Derivative (dy/dx): 2 cos(2x)
- Second Derivative (d²y/dx²): -4 sin(2x)