# Derivative Trick Rational Function

Hello Learners! Welcome to Crackexams99, In this post, I will tell you a differentiation trick that will be going to help you a lot while solving some questions.

What is the nth derivative of x^2log(3x) ?

If you have a rational function like this:

$\frac{d}{dx} \left( \frac{a(f(x))+b}{c(f(x))+d} \right)$

Then do cross multiply the numerator and square the denominator like this:

$=\frac{\begin{vmatrix} a & b \\ c & d \end{vmatrix}f’x}{(cf(x)+d)^2}$

For example:

$\frac{d}{dx} \left( \frac{4logx + 8}{-7logx+5} \right)$

By using the trick

$=\frac{(20+56)\frac{1}{x}}{(-7logx+5)^2}$
$=\frac{76}{x(-7logx+5)^2}$

That’s it.

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