Question

高数求偏导数

Answer 1

(1) z = arctan(y/x), ∂z/∂x = (-y/x^2)/[1+(y/x)^2] = -y/(x^2+y^2),
∂z/∂y = (1/x)/[1+(y/x)^2] = x/(x^2+y^2).
(2) z = √ln(xy), ∂z/∂x = [y/(xy)]/[2√ln(xy)] = 1/[2x√ln(xy)],
∂z/∂y = [x/(xy)]/[2√ln(xy)] = 1/[2y√ln(xy)]
(3) u = e^(xy^2z^3), ∂u/∂x = y^2z^3 e^(xy^2z^3),
∂u/∂y = 2xyz^3 e^(xy^2z^3), ∂u/∂z = 3xy^2z^2 e^(xy^2z^3)
(4) z = xln(xy), ∂z/∂x = ln(xy) + xy/(xy) = 1 + ln(xy)
∂^2z/∂x^2 = y/(xy) = 1/x, ∂^2z/∂x∂y = x/(xy) = 1/y
(5) z = ye^(2x)+xsin2y
∂z/∂x = 2ye^(2x) + sin2y, ∂z/∂y = e^(2x) + 2xcos2y
∂^2z/∂x^2 = 4ye^(2x), ∂^2z/∂x∂y = 2e^(2x) + 2cos2y,
∂^2z/∂y^2 = -4xsin2y