This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the partial derivatives for the given functions. i) For f(x,y) = x^3 + 5xy + 2y - 2: Step 1: Calculate the first partial derivatives. f_x(x,y) = ()/( x)(x^3 + 5xy + 2y - 2) = 3x^2 + 5y f_y(x,y) = ()/( y)(x^3 + 5xy + 2y - 2) = 5x + 2 Step 2: Calculate the second partial derivatives. f_xx(x,y) = ()/( x)(3x^2 + 5y) = 6x f_xy(x,y) = ()/( y)(3x^2 + 5y) = 5 f_yy(x,y) = ()/( y)(5x + 2) = 0 The derivatives for f(x,y) = x^3 + 5xy + 2y - 2 are: f_x(x,y) = 3x^2 + 5y f_y(x,y) = 5x + 2 f_xx(x,y) = 6x f_xy(x,y) = 5 f_yy(x,y) = 0 ii) For f(x,y) = 8 + 2x - xy^2: Step 3: Calculate the first partial derivatives. f_x(x,y) = ()/( x)(8 + 2x - xy^2) = -2 2x - y^2 f_y(x,y) = ()/( y)(8 + 2x - xy^2) = -2xy Step 4: Calculate the second partial derivatives. f_xx(x,y) = ()/( x)(-2 2x - y^2) = -4 2x f_xy(x,y) = ()/( y)(-2 2x - y^2) = -2y f_yy(x,y) = ()/( y)(-2xy) = -2x The derivatives for f(x,y) = 8 + 2x - xy^2 are: f_x(x,y) = -2 2x - y^2 f_y(x,y) = -2xy f_xx(x,y) = -4 2x f_xy(x,y) = -2y f_yy(x,y) = -2x Send me the next one 📸