Taking: f(x) = (x^(2)+(3*x)+1)/(x^(2)+(5*x)+8)

An application of the quotient rule of differentiation is required. This rule is given as:

Where g(x) = u(x)/v(x), g'(x) = ((u'(x)*v(x))-(v'(x) u(x)))/(v(x)^(2))*x)+8)is deconstructed to:

Hence for the case of the f(x) given, f(x) = (x^(2)+(3x)+1)/(x^(2)+(5

u(x) = x^(2)+(3

Hence:

(v(x))^2 = (x^(2)+(5

Applying a simple method of differentiation gives:

u'(x)= (2

Thus, bringing all the constituents together and entering them into the quotient rule formula:

f'(x) = ((u'(x)

Expanding and collecting like terms:

f'(x)=(2

This is as far as this expression can be simplified and hence the question has been answered fully.

Teachable points:Difference in degree is clearly -1 as is required by the definition of differentiation from first principlesThe f'(x) can be described as the rate of change of f(x) and can be used to quantify how f(x) varies as x variesFurther investigation into the graph of y=f(x) could occur from this, eventually allowing the plotting of this graph