Question

Question #0caf6

Answer 1

The curve has a slope of `-1` at `(2, 0)`.

Explanation:

The trick is to always remember you are differentiating with respect to `x` and to write "`dy/dx`" whenever you differentiate a different variable.

We differentiate the above equation using the power rule, which states that for a function `f(x) = x^n`, the derivative is given by `f'(x) = nx^(n - 1)`.

`2(1/2)x + 2y(dy/dx) + 2(dy/dx) - 0 = 0`

`x + 2y(dy/dx) + 2(dy/dx) = 0`

Solve for `dy/dx`

`2y(dy/dx) + 2(dy/dx) = -x`

`dy/dx(2y + 2) = -x`

`dy/dx = -x/(2y + 2)`

The slope of the curve at a point `x = a` can be obtained by evaluating `x= a` within `dy/dx`.

`dy/dx = -2/(2(0) +2)`

`dy/dx= -2/2`

`dy/dx= -1`

Hopefully this helps!