The Quotient Rule.





Quotient Rule is one of the Techniques of Differentiation.


The Quotient Rule states that:

The Derivative of  a Function “f(x)” divided by another function “g(x)” is the difference between the second function multiplied by derivative of first function and first function multiplied by derivative of second function whole divided by square of  the second function.

Mathematically we can write:

\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\textstyle \frac{g(x).f^|(x)-f(x).g^|(x)}{(g(x))^2}



Proof of Quotient Rule:

If, h(x)=f(x)/g(x)
and “a” is a fixed point then,


h^|(a)=\displaystyle\lim_{x\to a}\frac{h(x)-h(a)}{x-a}


=\displaystyle\lim_{x\to a}\frac{\frac{f(x)}{g(x)}-\frac{f(a)}{g(a)}}{x-a}


=\displaystyle\lim_{x\to a}\frac{g(a)f(x)-f(a)g(x)}{(x-a)g(x)g(a)}


=\displaystyle\lim_{x\to a}\frac{g(a)f(x)-g(a)f(a)+g(a)f(a)-f(a)g(x)}{(x-a)g(x)g(a)}


=
\displaystyle\lim_{x\to a}\left(\frac{1}{g(a).g(x)}\left(g(a).\frac{f(x)-f(a)}{x-a}-f(a).\frac{g(x)-g(a)}{x-a}\right)\right)


=\frac{1}{g(a).g(a)}[g(a).f^|(a)-f(a).g^|(a)]


=\textstyle \frac{g(a).f^|(a)-f(a).g^|(a)}{(g(a))^2}



Use of Quotient Rule:


Find the derivative of: \frac{4x^2+3}{3x^2-2}


Solution:
Let f(x)=4x^2+3 and g(x)=3x^2-2
Then,
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x).f^|(x)-f(x).g^|(x)}{(g(x))^2}


=\frac{(3x^2-2).8x-(4x^2+3).6x}{(3x^2-2)^2}


=\frac{-34x}{(3x^2-2)^2}



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