Derivative of (ax+b)/​(cx+d​)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = a x + b$ and $g{\left(x \right)} = c x + d$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $a x + b$ term by term:

      1. The derivative of the constant $b$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $a$

      The result is: $a$

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $c x + d$ term by term:

      1. The derivative of the constant $d$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $c$

      The result is: $c$

   Now plug in to the quotient rule:

   $\frac{a \left(c x + d\right) - c \left(a x + b\right)}{\left(c x + d\right)^{2}}$

The answer is:

$$\frac{a \left(c x + d\right) - c \left(a x + b\right)}{\left(c x + d\right)^{2}}$$