Rules of differentiation: Rules of computation for the derivative
The quotient rule for differentiation
Let #f# and #g# be functions. The quotient function #\dfrac{f}{g}# is the function that assigns the value #\dfrac{f(x)}{g(x)}# to #x#.
For example, if #f(x)=x+1# and #g(x)=x^3+1#, then the quotient function #\frac{f}{g}# has function rule
\[\frac{f}{g}(x) = \frac{f(x)}{g(x)}=\frac{x+1}{x^3+1}\tiny.\]
The following rule determines the derivative of a quotient function. Recall that #g^2# for a function #g# stands for #g\cdot g#.
Quotient rule for differentiation
Let #f# and #g# be differentiable functions. The derivative of #\dfrac{f}{g}# is #\dfrac{f'\cdot g-g'\cdot f}{g^2}#, so \[\left(\dfrac{f}{g}\right)'(x) = \dfrac{f'(x)\cdot g(x)-f(x) \cdot g'(x)}{g(x)^2}\tiny.\]
Let #h(x)=\dfrac{f(x)}{g(x)}#. We want to prove that #h'(x)=\dfrac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g(x)^2}#.
We will use the definition of #h# in the form #f(x)=g(x) \cdot h(x)#. The product rule then says that #f'(x)=g'(x) \cdot h(x) + g(x) \cdot h'(x)#. This means that: #h'(x)=\frac{f'(x)-g'(x)\cdot h(x)}{g(x)}#.
Since #h(x)=\dfrac{f(x)}{g(x)}#, we have #h'(x)=\frac{f'(x) \cdot g(x)-g'(x)\cdot f(x)}{g^2(x)}#.
So #h'(x)=\left(\dfrac{f}{g}\right)'(x)=\frac{f'(x) \cdot g(x)-f(x)\cdot g'(x)}{g^2(x)}#.
According to the quotient rule for differentiation we have\[\begin{array}{rcl}\dfrac{\dd}{\dd x}\left(\dfrac{f}{g}(x)\right) &=& \dfrac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g(x)^2}\\ &=& \dfrac{(2x+4)\cdot (2\cdot x+2)-2\cdot (x^2+4\cdot x-3)}{(2\cdot x+2)^2}\\ & =& \displaystyle {{x^2+2\cdot x+7}\over{2\cdot x^2+4\cdot x+2}} \end{array}\]
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