# Calculus

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26 Feb 2015
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### Calculus

• 1. * *NAME- PRAKHAR GUPTA *BRANCH- COMPUTER *DIVISION- CE-C *ROLL NO- 50 *ENROLLMENT NO- 140770107179 *SUBJECT- CALCULUS.PPT *SUBJECT CODE- 2110014 *TOPIC- DIFFERENCE BETWEEN ORDINARY DERIVATIVES AND PARTIAL DERIVATIVES *FACULTY NAME- PATEL DHRUVI
• 3. For the function y=f(x),the derivative of y with respect to x is denoted by dy/dx or yI or y1.This derivative is called Ordinary derivative as we differentiate the function of one independent variable only.
• 4. When the function involves two or more variables like u=f(x,y),then the derivative of u with respect to any one of the independent variables ,treating all other variables as constant is referred to as partial derivative of u with respect to that variable.
• 5. Let u=f(x,y) be a function of two independent variables x and y, then the first order partial derivative of u with respect to x is denoted by fx ,ux . ∂u/∂x(a,b)=limh→0 (f(a+h,b)−f(a,b))/h Let u=f(x,y) be a function of two independent variables x and y,then the first order partial derivative of u with respect to y is denoted by fy ,uy . ∂f/∂y(a,b)=limh→0 (f(a,b+h)−f(a,b))/h. *
• 6. • The difference between ordinary differentiation and partial differentiation lies in the idea of dependency of variables. • When taking an ordinary derivative (say dx/dy ) of a function of one or more variables, it must be assumed that all the variables depend on x. • For example, consider the function f( x, y) = (ln y) sin x + y2 -(1)
• 7. To compute df/dx we must treat the variable y as a function of x and use the chain-rule . Thus df/dx = (ln y) cos x + sin x (1/y) dx/dy + 2ydx/dy OR if computing df/dy we must treat x as a function of y df/dy =(1/y)sin x + (ln y) cos x dx/dy + 2y
• 8. • For partial derivatives it is instead assumed that all the variables are independent. Thus when applying ∂u/∂x to a function of several variables, we can assume that all variables apart from x are constants. • With the above example(1) we get ∂f/∂x = (ln y) cos x ∂f/∂y = (1/y) sin x + 2y
• 9. • There is a close connection between the two ideas and the chain rule. Partial derivatives are an important component of the computation of an ordinary derivative. • For a function of 3 variables f(x, y, z)the derivative df/dx can be computed by the formula df/dx =∂f/∂x +(∂f/∂y)(dy/dx) +(∂f/∂z)(dz/dx) • Similar formulae hold for functions of any number of variables.
• 10. • The concept of partial derivative and ordinary derivative are interrelated to each other. • The difference lies in dependency of variables. In Ordinary derivative the other variables depend on one variable with respect to which differentiation occurs and in Partial derivative all other variables are considered to be constant other than the one differentiating. *