Question

Question #42a53

Answer 1

`(dy)/(dx) = -46/27`

Explanation:

To find the slope of the curve, we will need to use implicit differentiation. We will differentiate every term, but in instances where we are taking the derivative of a `y` term, we must remember to multiply that derivative by `(dy)/(dx)` (often denoted as `y'` for convenience of writing).

Thus:

`32y^7 y' + 54x^5 = 5y' + 8`

Collect the `y'` terms on the left side of the equation, and move the remaining terms to the right side:

`32y^7 y' - 5y' = 8-54x^5`

Factor out the `y'`, and then divide by the factored out terms:

`y'(32y^7 - 5) = 8-54x^5`

`y' = (8-54x^5)/(32y^7-5)`

Now we evaluate this at the point `(1,1)` to find the slope:

`y' = (8-54)/(32-5) = (-46)/(27)`