Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = sin² x

Verified step by step guidance

1. Identify the function f(x) = sin²(x) and recognize that we need to find its derivative f'(x).
2. Use the chain rule to differentiate f(x). Recall that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x). Here, g(u) = u² and h(x) = sin(x).
3. Calculate the derivative of g(u) = u², which is g'(u) = 2u, and the derivative of h(x) = sin(x), which is h'(x) = cos(x).
4. Substitute h(x) back into g'(u) to find f'(x) = 2sin(x) * cos(x). This can also be expressed as f'(x) = sin(2x) using the double angle identity.
5. Express the relationship between the small change in x (dx) and the corresponding change in y (dy) using the formula dy = f'(x)dx, substituting the expression for f'(x) found in the previous step.