To differentiate the function sin(2x), we can use the chain rule from calculus, which is essential for differentiating composite functions.
First, let’s identify the outer and inner functions: in this case, the outer function is sin(u) and the inner function is u = 2x.
According to the chain rule:
if y = sin(u) and u = 2x, then
dy/dx = (dy/du) * (du/dx).
Now, we compute dy/du and du/dx:
- dy/du = cos(u), so dy/du = cos(2x).
- du/dx = 2, since the derivative of 2x is 2.
Now, we can combine these results:
dy/dx = cos(2x) * 2.
Thus, the derivative of sin(2x) is:
2 cos(2x).
This result shows how the chain rule allows us to differentiate functions that involve compositions of functions effectively.