What is the derivative of sqrt(a^2 - x^2) with respect to x?

• A. -x / sqrt(a^2 - x^2)
• B. x / sqrt(a^2 - x^2)
• C. -sqrt(a^2 - x^2)
• D. -a / sqrt(a^2 - x^2)

Answer: (A)

Using the chain rule on sqrt(u) with u = a^2 - x^2, differentiate step by step. The derivative of sqrt(u) is (1/(2 sqrt(u))) times du/dx. Here du/dx = -2x because a^2 is constant and the derivative of -x^2 is -2x. Multiply: (1/(2 sqrt(a^2 - x^2)))*(-2x) = -x / sqrt(a^2 - x^2). This is defined where a^2 - x^2 > 0, i.e., |x| < |a|. The negative sign comes from the inner derivative. Options that omit the -2x or treat the inner derivative as zero would not match this result.