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This is a mathematical problem and I am very bad at mathematics. I always get low marks in mathematics because I do not remember the formulas that use in maths and sometimes I forget the solution to a given problem. One of my friend from aws security specialist training is good at mathematic and he always get good marks in the and I always do prepratation from him when ever there is an exam of math. I will talk with him about your problem and hope he will solve this problem.
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These are functions, and only a Mathematician can help you out. If you find any professional, please ask him to suggest some essay writing service in Los Angeles for me. I have never taken any academic services before, but if someone suggests any brand, please tell me too. And you can get this query solved from Google assistant as well.
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
F(x,y)= f(w)
= 2w^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