Multivariable calculus

Second and mixed partials

Second and mixed partials, and why the mixed ones come out equal.

• Second Partial Derivatives of x^3 y^2Second partials of f(x,y) = x^3 y^2: f_xx = 6xy^2, f_xy = 6x^2 y, f_yy = 2x^3, with the Clairaut check that both mixed orders agree.
• Second Partial Derivatives of e^(xy)Second partials of f(x,y) = e^(xy): f_xx = y^2 e^(xy), f_yy = x^2 e^(xy), and the mixed partial f_xy = (1 + xy) e^(xy) from the product rule.
• Second Partials of x^3 - 3xy^2 and Why They CancelSecond partials of the monkey saddle x^3 - 3xy^2: f_xx = 6x, f_xy = -6y, f_yy = -6x. The pure ones sum to zero, so the function is harmonic.
• Second Partial Derivatives of ln(x^2 + y^2 + 1)Second partials of ln(x^2 + y^2 + 1) by the quotient rule, including the mixed partial -4xy over (x^2 + y^2 + 1)^2 and the Clairaut check.
• Second Partials of the Gaussian e^(-x^2 - y^2)Second partials of the Gaussian bump e^(-x^2-y^2): f_xx = (4x^2 - 2)E, f_xy = 4xyE, f_yy = (4y^2 - 2)E, plus the second derivative test at the peak.
• Second Partial Derivatives of sin(x)cos(y)Second partials of f(x,y) = sin(x)cos(y): f_xx = f_yy = -sin(x)cos(y) and f_xy = -cos(x)sin(y), with the eigenvalue relation Laplacian f = -2f.
• Second Partial Derivatives of arctan(xy)Second partials of arctan(xy) by the quotient rule: f_xx = -2xy^3/(1 + x^2y^2)^2 and the mixed partial (1 - x^2y^2)/(1 + x^2y^2)^2.
• Second Partial Derivatives of x/(1 + y^2)Second partials of x/(1 + y^2): f_xx = 0, f_xy = -2y/(1 + y^2)^2, f_yy = x(6y^2 - 2)/(1 + y^2)^3, and why a zero second partial is not flatness.
• Second Partial Derivatives of xy e^(x+y)Second partials of xy e^(x+y): f_xx = y(2 + x)E, f_xy = (1 + x)(1 + y)E, f_yy = x(2 + y)E, with the second derivative test at both critical points.
• Second Partial Derivatives of x^2 y z^3All six distinct second partials of f(x,y,z) = x^2 y z^3, including the three mixed ones, with Clairaut symmetry in three variables.
• Second Partials of x sin(y) + y cos(x)Second partials of x sin(y) + y cos(x): f_xx = -y cos(x), f_yy = -x sin(y), and the mixed partial cos(y) - sin(x), where both terms contribute.
• Second partial derivatives of e^x sin yFind f_xx, f_xy and f_yy for f(x,y) = e^x sin y, check Clairaut symmetry, and see why the Laplacian of this surface is exactly zero.
• Second partial derivatives of sin x cos yCompute f_xx, f_xy and f_yy for f(x,y) = sin x cos y, verify the mixed partials match, and use the Hessian to classify the point (pi/2, 0).
• Second partial derivatives of sin(xy)Work out f_xx, f_xy and f_yy for f(x,y) = sin(xy), where the mixed partial needs both the chain rule and the product rule.
• Second partials of the Gaussian e^(-x^2-y^2)Compute f_xx, f_xy and f_yy for the Gaussian bump f(x,y) = e^(-x^2-y^2), find the local maximum at the origin, and see where each curvature changes sign.
• Second partials of ln(1 + x^2 + y^2)Find f_xx, f_xy and f_yy for f(x,y) = ln(1 + x^2 + y^2) using the chain and quotient rules, and see why its Laplacian is always positive.
• Second partials of ln(e^x + e^y)Compute f_xx, f_xy and f_yy for the log-sum-exp function ln(e^x + e^y), and see why its Hessian determinant is exactly zero everywhere.
• Second partials of x e^y + y e^xCompute f_xx, f_xy and f_yy for f(x,y) = x e^y + y e^x, a clean demonstration that the two mixed partials agree even by very different routes.
• Second partials of sin(x + y^2)Find f_xx, f_xy and f_yy for f(x,y) = sin(x + y^2), where the y derivative needs the chain rule twice and then the product rule.