Find dy/dx in terms of x and y if (x−a)^9+y^9=a^9. Assume that a is a constant.
To find dy/dx in terms of x and y, we can differentiate both sides of the given equation implicitly using the chain rule.
Let's start by differentiating the equation (x−a)^9 + y^9 = a^9 with respect to x:
(d/dx) [(x-a)^9 + y^9] = (d/dx) (a^9)
Using the chain rule, we have:
9(x-a)^(9-1) * (d/dx) (x-a) + 9y^8 * (d/dx) y = 0
Simplifying, we have:
9(x-a)^8 * (d/dx) (x-a) + 9y^8 * (dy/dx) = 0
Now, let's isolate dy/dx:
9y^8 * (dy/dx) = -9(x-a)^8 * (d/dx) (x-a)
Dividing both sides by 9y^8, we get:
dy/dx = -(x-a)^8 * (d/dx) (x-a) / y^8
Now, let's simplify further:
The derivative of (x-a) with respect to x is simply 1.
Therefore, dy/dx = -(x-a)^8 / y^8.
So, the derivative dy/dx in terms of x and y for the given equation is -(x-a)^8 / y^8.