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Help, I need someone to teach me how to solve these questions!

Good morning, 
 
I need help, I'm studying for my calculus final exam and I'm having some troubles solving some questions we had in previous test. Please, can anyone teach me how to solve these?
 
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1)For 
 
z = ƒ (x,y) = √x2+ 3y
                    (root)
verify Clairaut's Theorem (ƒxy= ƒyx). 
 
 
 
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2) Find all partial derivatives for
g(p,q,r) = p.cos(πpq+qr2
                         (pi)
 
 
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I would really appreciate your help. 
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1 Answer

Both call for taking derivative of f with regard to x,y or g with regard to p,q,r. 
 
The first is x +3y. So fx is 1 and fxy is 0. Fy is 3 and fxy is 0. Qed!
 
The next I assume is p cos (pi*pq +qr^2) which we can use chain rulesubstituting u=pi*pq+qr^2.
Then g is p cos u where du/dp is pi*q, du/dq is pi*p+r^2, and du/dr is 2r
We use product rule to differentiate p cos u where y is p and z is cos u.
Then g/du is y *zu +z*yu (yuck!) = -psin u 
 
Now gp = g/du * du/dp which is -p sin (pi*pq+qr^2) * pi*q
And gq is g/du * du/dq and gr is g/du* du/dr.
 
Hope  that helped,
Deanna

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