Question

How do you implicitly differentiate `xy+2x+3x^2=-4`?

Answer 1

So, recall that for implicit differentiation, each term has to be differentiated with respect to a single variable, and that to differentiate some `f(y)` with respect to `x`, we utilise the chain rule:

`d/dx(f(y)) = f'(y)*dy/dx`

Thus, we state the equality:
`d/dx(xy) + d/dx(2x) + d/dx(3x^2) = d/dx(-4)`
`rArr x*dy/dx + y + 2 +6x = 0` (using the product rule to differentiate `xy`).

Now we just need to sort out this mess to get an equation `dy/dx =...`

`x*dy/dx = -6x-2-y`
`:. dy/dx = -(6x+2+y)/x` for all `x in RR` except zero.