Question

How do you find `(d^2y)/(dx^2)` for `5x^3=-4y^2+4`?

Answer 1

`(d^2y)/dx^2=-(240xy^2+225x^4)/(64y^3)`

Explanation:

First, we will use implicit differentiation to find `dy/dx`. Recall that implicit differentiation involves using the chain rule when differentiating a term with `y`.

Differentiating:

`15x^2=-8ydy/dx`

Solve for `dy/dx`:

`dy/dx=-(15x^2)/(8y)`

Differentiate again. This time the quotient rule will be used!

`(d^2y)/dx^2=-(30x(8y)-15x^2(8dy/dx))/(8y)^2`

Replace `dy/dx` with `-(15x^2)/(8y)`:

`(d^2y)/dx^2=-(30x(8y)-15x^2(8(-(15x^2)/(8y))))/(8y)^2`

Simplify:

`(d^2y)/dx^2=-(240xy-15x^2(-(15x^2)/y))/(64y^2)`

`(d^2y)/dx^2=-(240xy^2-15x^2(-15x^2))/(64y^3)`

`(d^2y)/dx^2=-(240xy^2+225x^4)/(64y^3)`