Partial Derivatives
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Partial derivatives are used to calculate the rate of change of a function of two or more variables with respect to one variable, while holding the other variables constant. The partial derivative of z with respect to x, denoted ∂z/∂x, is defined as the limit of the difference quotient as Δx approaches 0, while holding y constant. Similarly, the partial derivative of z with respect to y, denoted ∂z/∂y, is defined as the limit of the difference quotient as Δy approaches 0, while holding x constant. Notations for higher order partial derivatives are also introduced. An example problem finds the first and second order partial derivatives of the function z=x^2y^3+6
Partial Derivatives
- 2. Partial Derivatives BY:- Aman Singh R.no. 3 1st Sem Mechanical
- 3. What is Partial Derivative Let z = f(x,y) be function of two individual variables x and y the derivative with respect to x keeping y constant is called Partial derivative of z with respect to x. It is denoted by ∂z/∂x, ∂f/∂x, fx……… It is defined as ∂z/∂x = lim f(x+ ∂x ,y)- f(x,y)/ ∂x ∂x→0 x f
- 4. Contd…… Similarly the Partial Derivative of z w.r.t y keeping x as a constant is denoted by ∂z/∂y, ∂f/∂y, fy………. and it is defined as ∂z/∂y = lim f(x, y+ ∂y) – f(x, y)/ ∂y ∂y→0
- 5. Meaning of ∂f/∂x Partial Derivative of f w.r.t x provided that f is function of two or more than two variable. f ( ) ( ) xx yy f f f xy yx y x x y f f x y f f y x f y f y y f x f x x 2 2 2 2 2 2 ( ) ( )
- 6. Notations • ∂z/∂x = p ∂z/ ∂x = q • ∂2z/∂x2 = r ∂2z/∂x∂y = s • ∂2z/∂y2 = t
- 7. Example • z= x2y3 + 6x5ey find ∂z/∂x, ∂z/∂y, ∂2z/∂x∂y • ∂z/∂x = 2xy3 + 6x5ey • ∂z/∂y = 3x2y2 + x6ey • ∂2z/∂x∂y = ∂/∂x(∂z/∂y) = ∂/∂x (3x2y2 + x6ey ) = 6xy2 + 6x5ey