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

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

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

Answer: (A)

The chain rule is at work: when a function y depends on x through an inner function u(x), its derivative is dy/dx = (dy/du)·(du/dx). Here y = cos(u), so dy/du = -sin(u). Multiplying by du/dx, which is u', gives dy/dx = -sin(u)·u' = -u' sin(u). This is the derivative with respect to x. If the outer function were different, you’d get different forms: for example, with sin(u) you’d have u' cos(u); with tan(u) you’d have u' sec^2(u); with cot(u) you’d have -u' csc^2(u).

The chain rule is at work: when a function y depends on x through an inner function u(x), its derivative is dy/dx = (dy/du)·(du/dx). Here y = cos(u), so dy/du = -sin(u). Multiplying by du/dx, which is u', gives dy/dx = -sin(u)·u' = -u' sin(u). This is the derivative with respect to x. If the outer function were different, you’d get different forms: for example, with sin(u) you’d have u' cos(u); with tan(u) you’d have u' sec^2(u); with cot(u) you’d have -u' csc^2(u).