The equation of a curve is y =√(8x − x2). Find an expression for dy/dx and the coordinates of the stationary point on the curve,
We are given the equation of the curve:
`y = sqrt(8x - x^2)`
We want to solve for `(dy)/(dx)` . This could be done using the chain rule:
`(d(f(g(x))))/dx = f'(x)*g'(x)`
`(dy)/(dx) = [1/2 (-x^2 + 8x)^(-1/2) ] * (-2x + 8)`
This simplifies to:
`(-x + 4)/(sqrt(x^2 + 8x))` 
dy/dx = (-x + 4)/(sqrt(x^2 + 8x))
For the stationary point, we know that `(dy)/(dx) = 0` . Since we already know the derivative of the equation (equation 1), we simply equate it to 0.
`(-x + 4)/(sqrt(x^2 + 8x)) = 0`
This implies that `-x + 4 = 0` and `x = 4` .
To get the value of y, we use the original equation of the curve, and plug in this x value:
`y = sqrt(8x - x^2) = sqrt(8*4 - 4^2) = sqrt(32 - 16) = sqrt(16) = 4`
Hence, the stationary point (x,y) for which the derivative is 0 is the point (4, 4).