SOLUTION: Having trouble can not figure this out for f(x,y)=3x^2y^5+6x^6y^2 compute fxx(1,-1) = fxy(2,2)= fyy(-1,-1) = thanks so much for help

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Question 205262: Having trouble can not figure this out
for f(x,y)=3x^2y^5+6x^6y^2
compute
fxx(1,-1) =
fxy(2,2)=
fyy(-1,-1) =
thanks so much for help
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!
Having trouble can not figure this out
for f(x,y)=3x^2y^5+6x^6y^2
compute
f%5Bxx%5D(1,-1) =
f%5Bxy%5D(2,2) =
f%5Byy%5D(-1,-1) =
thanks so much for help


Find the first partials:

f%28x%2Cy%29=3x%5E2y%5E5%2B6x%5E6y%5E2

To find f%5Bx%5D(x,y), consider y to be
a constant, in the first term 3y%5E5 is considered
constant, and in the second term 6y%5E2 is considered
constant.  If you like, you can rewrite f%28x%2Cy%29 so what 
is to be held constant is in parentheses

f%28x%2Cy%29=%283y%5E5%29x%5E2%2B%286y%5E2%29x%5E6

Then we use the ordinary derivative formulas,
considering what is in parentheses as constant:

f%5Bx%5D%27%28x%2Cy%29%27=+2%283y%5E5%29x%2B6%286y%5E2%29x%5E5+

simplifying,

f%5Bx%5D%27%28x%2Cy%29%27=+6y%5E5x%2B36y%5E2x%5E5+

get it in alphabetical order

f%5Bx%5D%27%28x%2Cy%29%27=6xy%5E5%2B36x%5E5y%5E2

----------------

f%28x%2Cy%29=3x%5E2y%5E5%2B6x%5E6y%5E2

To find f%5By%5D(x,y), consider x to be
a constant, in the first term 3x%5E2 is considered
constant, and in the second term 6x%5E6 is considered
constant.  If you like, you can rewrite f%28x%2Cy%29 so what 
is to be held constant is in parentheses

f%28x%2Cy%29=%283x%5E2%29y%5E5%2B%286x%5E6%29y%5E2

Then we use the ordinary derivative formulas,
considering what is in parentheses as constant:

f%5By%5D%27%28x%2Cy%29%27=+5%283x%5E2%29y%5E4%2B2%286x%5E6%29y+

simplifying,

f%5By%5D%27%28x%2Cy%29%27=15x%5E2y%5E4%2B12x%5E6y+

===============================================

Find the second partial derivative f%5Bxx%5D, which
is the partial derivative with respect to x of 
the partial derivative with respect to x.

Start with the partial derivative with respect to x.

f%5Bx%5D%27%28x%2Cy%29%27=6xy%5E5%2B36x%5E5y%5E2

To find f%5Bxx%5D(x,y), consider y to be
a constant, in the first term 6y%5E5 is considered
constant, and in the second term 36y%5E2 is considered
constant.  If you like, you can rewrite f%5Bx%5D so what 
is to be held constant is in parentheses

f%5Bx%5D%27%28x%2Cy%29%27=%286y%5E5%29x%2B%2836y%5E2%29x%5E5

Then we use the ordinary derivative formulas,
considering what is in parentheses as constant:

f%5Bxx%5D%27%28x%2Cy%29%27=+%286y%5E5%29%2B5%2836y%5E2%29x%5E4+

simplifying,

f%5Bxx%5D%27%28x%2Cy%29%27=+6y%5E5%2B180y%5E2x%5E4+

get it in alphabetical order

f%5Bxx%5D%27%28x%2Cy%29%27=6y%5E5%2B180x%5E4y%5E2

And substituting

f%5Bxx%5D%27%281%2C-1%29%27=6%28-1%29%5E5%2B180%281%29%5E4%28-1%29%5E2

f%5Bxx%5D%27%281%2C-1%29%27=6%28-1%29%2B180%281%29%281%29

f%5Bxx%5D%27%281%2C-1%29%27=-6%2B180

f%5Bxx%5D%27%281%2C-1%29%27=174

=============================================== 

===============================================

Find the second partial derivative f%5Bxy%5D, which
is the partial derivative with respect to y of 
the partial derivative with respect to x.

Start with the partial derivative with respect to x.

f%5Bx%5D%27%28x%2Cy%29%27=6xy%5E5%2B36x%5E5y%5E2

To find f%5Bxy%5D(x,y), consider x to be
a constant, in the first term 6x is considered
constant, and in the second term 36x%5E5 is considered
constant.  If you like, you can rewrite f%5Bx%5D so what 
is to be held constant is in parentheses

f%5Bx%5D%27%28x%2Cy%29%27=%286x%29y%5E5%2B%2836x%5E5%29y%5E2

Then we use the ordinary derivative formulas,
considering what is in parentheses as constant:

f%5Bxy%5D%27%28x%2Cy%29%27=5%286x%29y%5E4%2B2%2836x%5E5%29y+

simplifying,

f%5Bxy%5D%27%28x%2Cy%29%27=30xy%5E4%2B72x%5E5y+

Substituting:

f%5Bxy%5D%27%282%2C2%29%27=30%282%29%282%29%5E4%2B72%282%29%5E5%282%29+

f%5Bxy%5D%27%282%2C2%29%27=30%282%29%2816%29%2B72%2832%29%282%29+

f%5Bxy%5D%27%282%2C2%29%27=960%2B4608+

f%5Bxy%5D%27%282%2C2%29%27=5568

=============================================== 

f%5By%5D%27%28x%2Cy%29%27=15x%5E2y%5E4%2B12x%5E6y+

Find the second partial derivative f%5Byy%5D, which
is the partial derivative with respect to y of 
the partial derivative with respect to y.

Start with the partial derivative with respect to y.

f%5By%5D%27%28x%2Cy%29%27=15x%5E2y%5E4%2B12x%5E6y

To find f%5Byy%5D(x,y), consider x to be
a constant, in the first term 15x%5E2 is considered
constant, and in the second term 12x%5E6 is considered
constant.  If you like, you can rewrite f%5By%5D so what 
is to be held constant is in parentheses

f%5By%5D%27%28x%2Cy%29%27=%2815x%5E2%29y%5E4%2B%2812x%5E6%29y

Then we use the ordinary derivative formulas,
considering what is in parentheses as constant:

f%5Byy%5D%27%28x%2Cy%29%27=4%2815x%5E2%29y%5E3%2B%2812x%5E6%29+

simplifying,

f%5Byy%5D%27%28x%2Cy%29%27=50x%5E2y%5E3%2B12x%5E6+

Substituting:

f%5Byy%5D%27%28-1%2C-1%29%27=50%28-1%29%5E2%28-1%29%5E3%2B12%28-1%29%5E6+

f%5Byy%5D%27%282%2C2%29%27=50%281%29%28-1%29%2B12%281%29+

f%5Byy%5D%27%282%2C2%29%27=-50%2B12+

f%5Byy%5D%27%282%2C2%29%27=-38

Edwin