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.
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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.
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