Techniques of Differentiation - Exercise 1

Exercise 1. Find the derivative of the function

\begin{displaymath}f(x) = \frac{ax+b}{cx+d}\cdot\end{displaymath}

Is there a nice way to rewrite this derivative?

Answer. We use the quotient rule. We have

\begin{displaymath}f'(x) = \frac{a(cx+d) - c(ax+b)}{(cx+d)^2}\end{displaymath}

which gives

\begin{displaymath}f'(x) = \frac{ad - cb}{(cx+d)^2}\;\cdot\end{displaymath}

Another way to write the formula is

\begin{displaymath}f'(x) = \frac{\left\vert\begin{array}{lr} a & b \\ c & d \end{array}\right\vert}{(cx+d)^2}\;\cdot\end{displaymath}

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