Let y be defined implicitly by the equation ln(8y)=5xy.

Question

A) Use implicit differentiation to find the second derivative of y with respect to x. ((d^2)y)/(d(x^2)= ?

Note: Your answer should only involve the variables x and y.

You should simplify your answer as much as possible.

B) Find the point on the curve where ((d^2)y)/(d(x^2)=0.

Answer 1

I'm going to define y' as dy/dx, now lets start by taking the first derivative:

8y'/(8y)=5y+5xy', y'-5xyy'=5y², y'(1-5xy)=5y², y'=5y²/(1-5xy)

now lets let y'' be d²y/(dx²) and take the second derivative:

y''=((1-5xy)(10yy')-(5y²)(5y+5xy'))/(1-5xy)²

Now wherever you see a y', replace it with the equation we had above y'=5y²/(1-5xy):

y''=((1-5xy)(10y(5y²/(1-5xy)))-(5y²)(5y+5x(5y²/(1-5xy))))/(1-5xy)² (I will leave the simplifying to you)

there is your second derivative

now for part b) set y''=0 and solve for the values of x and y that satisfy that equation