Definition of Derivative: f'(x) =

_{lim h-->0}(f(x+h)-f(x)) / h

Problem: find p'(x): p(x) = (3x+1)/(7x-6)

Work: (I would list all the algebra in detail but it's intensive and the answer is correct, per the back of the book)

Result: p'(x) = -25/(7x-6)

^{2}

The product rule: if F(x) = f(x)g(x), then: F'(x) = f(x)g'(x) + f'(x)g(x)

The power rule: if f(x) = x

^{n}, then: f'(x) = nx

^{n-1}

Problem: find p'(x): p(x) = (3x+1)/(7x-6)

Work:

for the product rule, let F(x) = p(x), f(x) = (3x+1) and g(x) = 1/(7x-6)

rewrite f(x) and g(x) as power expressions: let f(x) = (3x+1)

^{1}and g(x) = (7x-6)

^{-1}

per the power rule, f'(x) = 1(3x+1)

^{0}and g'(x) = -1(7x-6)

^{-2}

per the product rule: p'(x) =

= (3x+1)

^{1}* -1(7x-6)

^{-2}+ 1(3x+1)

^{0}* (7x-6)

^{-1}

= -(3x+1)/(7x-6)

^{2}+ 1/(7x-6)

= -(3x+1)*(7x-6) + (7x-6)

^{2}/ (7x-6)

^{3}

= -(3x+1) + (7x-6) / (7x-6)

^{2}

= 4x-7 / (7x-6)

^{2}

Result = 4x-7 / (7x-6)

^{2}

This result is clearly different than the answer found using the definition of the derivative. Please help! Thanks...