If y = sec(u) and u = x^2, dy/dx equals?

If y = sec(u) and u = x^2, dy/dx equals?

• A. x sec(x^2) tan(x^2)
• B. 2 sec(x^2) tan(x^2)
• C. 2x sec(x^2) tan(x^2)
• D. 2x sec^2(x^2)

Answer: (C)

The derivative uses the chain rule. For y = sec(u) with u = x^2, differentiate sec with respect to its argument and multiply by the derivative of the inner function: d/dx[sec(u)] = sec(u) tan(u) · du/dx. Here du/dx = 2x and u = x^2, so dy/dx = sec(x^2) tan(x^2) · 2x = 2x sec(x^2) tan(x^2). This is the correct form because it combines the outer derivative sec(u) tan(u) with the inner derivative 2x from u = x^2. The other possibilities would be missing the 2x, missing the tan factor, or using sec^2, which corresponds to a different derivative (not of sec).

The derivative uses the chain rule. For y = sec(u) with u = x^2, differentiate sec with respect to its argument and multiply by the derivative of the inner function: d/dx[sec(u)] = sec(u) tan(u) · du/dx. Here du/dx = 2x and u = x^2, so

dy/dx = sec(x^2) tan(x^2) · 2x = 2x sec(x^2) tan(x^2).

This is the correct form because it combines the outer derivative sec(u) tan(u) with the inner derivative 2x from u = x^2. The other possibilities would be missing the 2x, missing the tan factor, or using sec^2, which corresponds to a different derivative (not of sec).