Question

total derivative of the function xy is equal to​

Answer 1

THE TOTAL DERIVATIVE OF xy IS = dx. y + dy . x

• In total derivative  , the function is differentiated with respect to all the variables present in function i.e x , y , z.

• To differentiate the product of two functions we use product  rule. the product rule is given by differentiating first function and keeping second function as it is and then differentiating second function and keeping the first function as it is .

• d(a.b)= d(a) . b + d(b). a -------------> product rule

• hence total derivative of x.y is given by

• d(x.y)= dx . y + dy . x  -------------------> ANS

thanks!!

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

TOTAL DERIVATIVE

Let z = f(x, y) be a function of two independent variables x and y defined in a domain

be a function of two independent variables x and y defined in a domain N

Also let it be differentiable at a point (x,y) on the domain N

Then The total differential of z at (x,y) denoted by dz and defined as

$\displaystyle \: \sf{dz = \frac{ \delta z}{ \delta x}dx + \frac{ \delta z}{ \delta y}dy} \:$

TO DETERMINE

The total derivative of the function xy

CALCULATION

$\sf{Let \: \: \: z = f(x, y) = xy}$

So

$\displaystyle \: \sf{ \frac{ \delta z}{ \delta x} = y } \:$

$\displaystyle \: \sf{ \frac{ \delta z}{ \delta y} = x} \:$

So the required total differential is

$\displaystyle \: \sf{dz = \frac{ \delta z}{ \delta x}dx + \frac{ \delta z}{ \delta y}dy} \:$

$\implies \: \displaystyle \: \sf{dz = y \: dx + x \: dy} \:$

RESULT

The required answer is

$\boxed{ \: \displaystyle \: \sf{dz = y \: dx + x \: dy} \: \: } \:$

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