Question

How do you differentiate implicitly to find the slope of the curve `y^4 + x^3 = y^2 + 10x` at the given point (0,1)?

Answer 1

`y^4 + x^3 = y^2 + 10x`

Differentiate both sides of the equation with respect to `x`.

Use whichever notation you prefer:

`D_x(y^4 + x^3) = D_x(y^2 + 10x)`

`d/dx(y^4 + x^3) =d/dx( y^2 + 10x)`

`4y^3 dy/dx +3x^2 = 2y dy/dx +10`

Now, if we want the general formula for `dy/dx`, solve algebraically for `dy/dx`, but all we have been aksed for is `dy/dx` when `x=0` and `y=1`, so let's just do that:

`4(1)^3 dy/dx +3(0)^2 = 2(1) dy/dx +10`

`4 dy/dx = 2 dy/dx + 10`

`dy/dx = 10/2 =5`