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Punjabi to English: Math's General field: Other Detailed field: Mathematics & Statistics
Source text - Punjabi welcome to the lectures on engineering mathematics one and this is lecture number ten we will be talking about partial derivatives of higher order
in the last lecture what we have seen the partial derivatives of f
basically the first order partial derivatives we have done in the last lecture so what was it was the fundamental definition of the partial derivative with respect to x at the point x zero y zero
the increment in x zero because we are taking or we are talking about the partial derivative with respect to x and the function value divided by the delta x increment and taking this limit if this limit exists
then we have the partial derivative with respect to x the first order partial derivative and similarly we have the first order partial derivative with respect to y at this point x zero y zero so when this limit exist
now we will continue this for higher order derivative so what will happen if we take for example the derivative of f with respect to x two times
in this case this is the notation so we have the partial derivative with respect to x and for example we want to take again the partial derivative with respect to x so the idea is
idea remains the same because we have this partial derivative with respect to x so this is another function and then we are taking partial derivative with respect to x of this function so basically this is like
del over del x and then we have some other function which am calling as f x which is basically the notation of this a partial derivatives so what we are now doing we are taking the partial derivatives of this function f x
and now the same definition what we have learnt already that the delta x goes to zero we will be talking about this limit so our function here f x and for example if we take at x not y not points so here with respect to x means you will make in increment here
in x and then y not will remain and then we have the f x at x not and y not and divided by this delta x so that will be the definition now of the second order derivative here with respect to x
so here the notation again we instead of this we also use f xx the derivative of f two times with respect to x or we also use this notation del two f over del x square so again the second order derivative a partial or derivative of f with respect to x
similarly we have the notation when we take the partial derivative of x and then we are taking the partial derivative with respect to y
in this case we use this notation f y x or del two f over del y del x in some literature people use this as f x y or some other way around del x del y
so will be following this notation the partial derivative with respect to x and then we take once again the partial derivative with respect to y so we will keep this order
here y x and f y x and del two f del y del x this means the first three partial derivative with respect to x and then with respect to y
so similar notation for the two times partial derivative with respect to y
which we will denote as here with respect to x here f y y or del two f over del y two
or we can have like first the partial derivative with respect to y and then we take the partial derivative with respect to x
so this we will denote as f x y or del two f del x del y so this is just the notation and these derivatives here f x y or f y x are called the mixed derivatives because here we have used both the variables x and then with respect to y or first with respect to y then with respect to x
so these are called the mixed derivatives
let us directly compute this mixed derivative del two f over del x del y and del two f over del y del x at the origin of this function
as we have discussed the del two f over del x del y means that the partial derivative with respect to x of the partial derivative of with respect to y
in short we can denote this like f y so we have partial derivative f y and with respect to x so we want to
get the partial derivative of this f y with respect to x so note that this f y is also a function of x and y so we can talk about this partial derivative with respect to x so as per the definition
again with respect to x so a partial derivative with respect to x so we will take the limit delta x goes to zero f y the function and
we are talking about the derivative at a origin so zero zero you have zero plus delta x which is delta x zero minus f y and at zero zero so now for
the function we have use f y so f y delta x zero f y zero zero o delta x the same definition of the partial derivatives with respect to x so instead of f we have used here f y
because we are taking the derivative with respect to x of the function f y
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