Consider the curve y=x/(x+4)^0.5. (i) Show that the derivative of the curve is given by dy/dx= (x+8)/2(x+4)^3/2 and (ii) hence find the coordinates of the intersection between the left vertical asymptote and the line tangent to the curve at the origin.

The unsimplified form of the derivative can be obtained fairly easily with use of the quotient rule. The trick for simplification is to multiply top and bottom by (x+4)^0.5, this allows manipulation of the numerator into the correct form and almost gives the final form of the denominator. The second part consists of three stages, first the equation of the asymptote line must be found; By inspection it should be clear that as x approaches -4 the denominator of the original curve equation approaches zero, hence the asymptote is at x=-4. The second stage is to find the tangent line at the origin, this means substituting x=0 into the derivative in order to find the gradient (dy/dx=1/2 here). Then the equation for the tangent can be seen to be y=x/2. Finally evaluate y=x/2 at x=-4 to arrive at the answer, (-4, -2).