F(x, y) = f(xy^2 + x^2 )
I need to find :
dF/dx and dF/dy
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dF/dx = d/dx[f(xy^2 + x^2)] = f'(xy^2 + x^2)*d(xy^2 + x^2/dx = f'(xy^2 + x^2) * (y^2 + 2x)
Not knowing anything about f this is the best we can do.
We can write this using a substitution by letting w = xy^2 + x^2:
so now F(x,y) = f(w) and by the chain rule dF/dx =d[ f(w)]/dx = f'(w) * dw/dx
which is more compact way of writing it.
It might be easier to see how this works if we use a specific example for f(w), say some thing easy like f(w) =2w^2 where w = xy^2 + x^2
Now we have 2 choices to proceed and will get the same answer both ways:
1) dF/dx = f'(w) * dw/dx
= [4w]*[y^2 + 2x] --- holding y constant
= [4xy^2 + 4x^2]*[y^2 + 2x]
= 4xy^4 + 8x^2y^2 + 4x^2y^2 + 8x^3
= 4xy^4 + 12x^2y^2 + 8x^3
2) Find F in terms of x and y first and then differentiate:
f(w) = 2w^2 where w = xy^2 + x^2
= 2[xy^2 + x^2]^2
= 2[x^2y^4 + 2x^3y^2 + x^4]
= 2x^2y^4 + 4x^3y^2 + 2x^4
Keeping y constant and differentiating wrt x:
dF/dx = 4xy^4 + 12x^2y^2 + 8x^3
Same for dF/dy
Thank you very much .