Let y=f(x) be a twice-differentiable function such that f(−1)=5 and dy/dx=1/5(xy^2+4y)^2 . What is the value of d^2y/dx^2 at x=−1 ?

Answer 1

The fact that f(-1) = 5 just tells you than when x = -1, y=5.

Evaluate dy/dx at x = -1:

y'(-1,5) = 1/5[(-1)(25) +20]² = 5.

Now find y'' from y' using implicit differentiation.

y'' = 1/5 (2) (2xyy' + y² + 4y')

In the above substitute x -1, y = 5, y' = 5 and evaluate.