Questions: Differentiate. f(x)=(ax+b)/(cx+d) f'(x)=

Solution Steps

Let \( f(x) = \frac{a x + b}{c x + d} \).

To differentiate \( f(x) \), we apply the quotient rule, which states that if \( f(x) = \frac{u(x)}{v(x)} \), then

\[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \]

where \( u(x) = a x + b \) and \( v(x) = c x + d \).

Calculate the derivatives:

• \( u'(x) = a \)
• \( v'(x) = c \)

Substituting into the quotient rule gives:

\[ f'(x) = \frac{a(c x + d) - (a x + b)c}{(c x + d)^2} \]

Simplifying the numerator:

\[ f'(x) = \frac{a(c x + d) - c(a x + b)}{(c x + d)^2} \]

This results in:

\[ f'(x) = \frac{a(c x + d) - c(a x + b)}{(c x + d)^2} = \frac{a}{c x + d} - \frac{c(a x + b)}{(c x + d)^2} \]

Final Answer