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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