If y = tan(u), dy/dx equals?

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

Answer: (C)

Use the chain rule. If y = tan(u(x)), the derivative with respect to x is the derivative of tan at the inner function times the derivative of the inner function. The derivative of tan(s) with respect to s is sec^2(s), so by the chain rule you get dy/dx = sec^2(u(x)) · du/dx. Writing du/dx as u', this is dy/dx = u' sec^2(u). This also matches special cases: if u is constant, dy/dx is 0; if u = x, dy/dx = sec^2(x).