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.
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