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Given f(x) = (x^(2)+(3*x)+1)/(x^(2)+(5*x)+8), find f'(x) and simplify your answer.

Taking: f(x) = (x^(2)+(3x)+1)/(x^(2)+(5x)+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))
Hence for the case of the f(x) given, f(x) = (x^(2)+(3
x)+1)/(x^(2)+(5
x)+8)is deconstructed to:
u(x) = x^(2)+(3x)+1v(x)=x^(2)+(5x)+8
Hence:
(v(x))^2 = (x^(2)+(5x)+8)^2
Applying a simple method of differentiation gives:
u'(x)= (2
x)+3v'(x)= (2x)+5
Thus, bringing all the constituents together and entering them into the quotient rule formula:
f'(x) = ((u'(x)v(x))-(v'(x)u(x)))/(v(x)^(2))f'(x)= ((((2x)+3)(x^(2)+(5
x)+8))-(((2x)+5)x^(2)+(3x)+1))/(x^(2)+(5x)+8)^2)
Expanding and collecting like terms:
f'(x)=(2x^(2)+(14x)+19)/((x^(2)+(3*x)+1)^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

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