# 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

jeew-m | Certified Educator

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`