How do you find the slope of the line normal to the curve ##(x^2+4)*y=8##?

The normal line is perpendicular to the tangent line, meaning that it will have the opposite reciprocal slope of the tangent line. We use the derivative to give us the slope of the tangent line.

1) Find the derivative:

The function ##(x^2+4)*y=8## is written implicitly. We could therefore use to find the derivative. However, it is fairly simple to solve the function for y explicitly, which avoids the need for implicit differentiation. We shall take that approach.

Divide both sides of the equation by ##(x^2+4)## and we obtain the equation:

##y = 8/(x^2+4)## which is equivalent to ##y= 8*(x^2+4)^(-1)##. We can now use the to differentiate.

##y'=-1(x^2+4)^(-2)*2x## or ##y'=(-2x)/(x^2+4)^2##.

2) Find the slope of the tangent line.

At whatever given point (for example, x= 2) we wished to find the slope of the normal line, we would begin by finding the slope of the tangent line.

Using x=2 as our example, ##y'(2)=(-2*2)/(2^2+4)^2 = -4/8^2 =-4/64=-1/16##.

3) Find the slope of the normal line.

Since the slope of the tangent line at x=2 is ##-1/16##, the slope of the normal line would be the opposite reciprocal of this number or 16.

The same procedure could now be used for any other constant.