Find dy/dx in terms of x and y if (x−a)^9+y^9=a^9. Assume that a is a constant
9(x-a)^8 + 9y^8 y'=0
solve for y'
To find dy/dx in terms of x and y, we need to differentiate the given equation with respect to x.
First, let's rewrite the equation as:
(x−a)^9 + y^9 = a^9
Now, we'll differentiate both sides of the equation with respect to x using the chain rule.
Differentiating the left side:
Using the chain rule, we get:
d/dx [(x−a)^9] + d/dx[y^9] = d/dx[a^9]
To differentiate (x−a)^9, we'll use the chain rule again and the power rule. Let u = x−a, so we have:
d/dx [(x−a)^9] = d/du [u^9] * d/dx (x−a)
= 9*u^8 * (d/dx x)
= 9(x−a)^8
To differentiate y^9 with respect to x, we use the chain rule and the fact that y is a function of x. So:
d/dx[y^9] = d/dy [y^9] * d/dx y
= 9y^8 * (d/dx y)
Finally, to differentiate a^9 with respect to x, we need to consider a as a constant. So:
d/dx [a^9] = 0, since a^9 is a constant.
Now, we can substitute these derivatives back into the equation:
9(x−a)^8 + 9y^8 * (d/dx y) = 0
Since we want to find dy/dx, we can isolate the term (d/dx y):
9y^8 * (d/dx y) = -9(x−a)^8
Dividing both sides by 9y^8:
d/dx y = -(x−a)^8 / y^8
Therefore, dy/dx in terms of x and y is given by:
dy/dx = -(x−a)^8 / y^8