How should you calculate partial derivative ∂f/∂x using first the chain rule & directly in MAPLE?

Question

If f(u,v) = vsinu+v^2, u(x,y)=tan^−1(y/x), v=sqrt(x^2 + y^2)

Calculate using chain rule and directly substituting for u(x,y), v(x,y).

Answer 1

Your syntax is not altogether clear to me, but I am guessing you are trying to achieve something like this:

uveqs := [u(x,y)=arctan(y/x), v(x,y)=sqrt(x^2+y^2)]:
f := v(x,y)*sin(u(x,y))+v(x,y)^2:

Differentiate f with respect to x (noticing the effect of the chain-rule),

K := diff(f,x):

      diff(v(x,y),x) * sin(u(x,y))
      + v(x,y) * diff(u(x,y),x) * cos(u(x,y))
      + 2*v(x,y) * diff(v(x,y),x)

Now substitute for u(x,y) and v(x,u), and simplify,

S1 := eval(K,uveqs):
simplify(S1);

                 2 x

Alternatively, first substitute for u(x,y) and v(x,y) into f itself,

fxy := eval(f,uveqs):

    (x^2+y^2)^(1/2)*y/x/(1+y^2/x^2)^(1/2)+x^2+y^2

And then differentiate with respect to x, and simplify,

S2 := diff(fxy,x):
simplify(S2);

                 2 x

If you have trouble understanding the mechanisms, you might also compare these two pairs of operations,

diff( v(x,y), x );
eval( %, uveqs );

and,

eval( v(x,y), uveqs );
diff( %, x );