Find all second partial derivatives, `f_(x x)(x,y),f_(xy)(x,y),f_(yx)(x,y) and f_(yy)(x,y)` a)`f(x,y)=ln(5x^2-7y^3)`   b)`f(x,y)=x^2y^3+e^(2x+3y)`xx,xy,yx,and yy are all small  e is to the power...

Find all second partial derivatives, `f_(x x)(x,y),f_(xy)(x,y),f_(yx)(x,y) and f_(yy)(x,y)`

a)`f(x,y)=ln(5x^2-7y^3)`

 

b)`f(x,y)=x^2y^3+e^(2x+3y)`

xx,xy,yx,and yy are all small 

e is to the power of all parts of 2x+3y

Asked on by jjtyme

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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Since the editors are allowed to answer one question at a time i will only do the first part.

 

In partial derivatives when differentiating w.r.t x y is considered as constant and vice versa.

f(x,y) = `ln(5x^2-7y^3)`

fx = `(delf)/(delx)`

fx = `[1/(5x^2-7y^3)]*[10x]`

fxx =

      = `(5x^2-7y^3)*10-10x*(10x)/[(5x^2-7y^3)^2]`

      = `-(50x^2+70y^3)/[(5x^2-7y^3)^2]`

 

fxy = `(delfx)/(dely)`

      = `-10x(-21y^2)/[(5x^2-7y^3)^2]`

      = `(210xy^2)/[(5x^2-7y^3)^2]`

     


fy = `(delf)/(dely)`

    = `[1/(5x^2-7y^3)]*[-21y^2]`

    = `(-21y^2)/(5x^2-7y^3)`

 

fyy = `(delfy)/(dely)`

     =  `[(5x^2-7y^3)*(-42y)+21y^2(-21y)]/(5x^2-7y^3)^2`

 

fyx = `(delfy)/(delx)`

     = `21y^2(10x)/(5x^2-7y^3)^2`

     = `210(xy^2)/(5x^2-7y^3)^2`

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