Partial Derivatives Ho Xuan Binh

If f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. If f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. If f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant.

Natural Science Department – Duy Tan University Partial Derivatives In this section, we will learn: Various aspects of partial derivatives Lecturer: Ho Xuan Binh Da Nang-01/2015 Natural Science Department – Duy Tan University PARTIAL DERIVATIVES If f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant Then, we are really considering a function of a single variable x: g(x) = f(x, b) Partial Derivatives   Natural Science Department – Duy Tan University PARTIAL DERIVATIVES If g has a derivative at a, we call it the partial derivative of f with respect to x at (a, b) We denote it by: fx(a, b) f ( a + h, b ) − f ( a , b ) f x (a, b) = lim h →0 h Partial Derivatives   Natural Science Department – Duy Tan University PARTIAL DERIVATIVES Similarly, the partial derivative of f with respect to y at (a, b), denoted by fy(a, b), is obtained by: f ( a, b + h) − f ( a, b) f y (a, b) = lim h →0 h Partial Derivatives   Natural Science Department – Duy Tan University PARTIAL DERIVATIVES If f is a function of two variables, its partial derivatives are the functions fx and fy defined by: f ( x + h, y ) − f ( x , y ) f x ( x, y ) = lim h →0 h f ( x, y + h ) − f ( x , y ) f y ( x, y ) = lim h →0 h Partial Derivatives   Natural Science Department – Duy Tan University NOTATIONS FOR PARTIAL DERIVATIVES If z = f(x, y), we write: ∂f ∂ ∂z f x ( x, y ) = f x = = f ( x, y ) = ∂x ∂x ∂x = f1 = D1 f = Dx f ∂f ∂ ∂z f y ( x, y ) = f y = = f ( x, y ) = ∂y ∂y ∂y = f = D2 f = Dy f Partial Derivatives   Natural Science Department – Duy Tan University RULE TO FIND PARTIAL DERIVATIVES OF z = f(x, y) To find fx, regard y as a constant and differentiate f(x, y) with respect to x To find fy, regard x as a constant and differentiate f(x, y) with respect to y Partial Derivatives   Natural Science Department – Duy Tan University Example If f(x, y) = x3 + x2y3 – 2y2 find fx(2, 1) and fy(2, 1) Partial Derivatives   Natural Science Department – Duy Tan University If Example  x  f ( x, y ) = sin  ÷ 1+ y   calculate ∂f ∂x ∂f and ∂y Partial Derivatives   LOGO Thank you for your attention