If y = sin(u), where u is a differentiable function of x, what is dy/dx?

If y = sin(u), where u is a differentiable function of x, what is dy/dx?

• A. u' cos(u)
• B. -u' sin(u)
• C. u' sec^2(u)
• D. -u' csc^2(u)

Answer: (A)

When a function is a composition y = sin(u(x)), use the chain rule: dy/dx = (dy/du) · (du/dx). Here dy/du is cos(u), and du/dx is u'. So dy/dx = cos(u) · u' = u' cos(u). This is the correct form because it applies the derivative of the outer function with respect to its inner argument and then multiplies by how the inner argument changes with x. The other expressions would come from differentiating different outer functions (for example, the derivative of cos(u) is -sin(u) · du/dx, and derivatives like sec^2(u) or csc^2(u) come from tan(u) or cot(u), not sin(u)).

When a function is a composition y = sin(u(x)), use the chain rule: dy/dx = (dy/du) · (du/dx). Here dy/du is cos(u), and du/dx is u'. So dy/dx = cos(u) · u' = u' cos(u). This is the correct form because it applies the derivative of the outer function with respect to its inner argument and then multiplies by how the inner argument changes with x. The other expressions would come from differentiating different outer functions (for example, the derivative of cos(u) is -sin(u) · du/dx, and derivatives like sec^2(u) or csc^2(u) come from tan(u) or cot(u), not sin(u)).