Question

find the general derivative ( f(x) ) for ( f(x) = \frac{12}{x+3} ) using the alternative formula for the derivative.

Explanation:

Step1: Recall the alternative derivative formula

The alternative formula for the derivative of a function \( f(x) \) is \( f^{\prime}(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h} \). First, we find \( f(x + h) \) for \( f(x)=\frac{12}{x + 3} \).

\( f(x + h)=\frac{12}{(x + h)+3} \)

Step2: Substitute into the derivative formula

Substitute \( f(x + h) \) and \( f(x) \) into the formula:

\( f^{\prime}(x)=\lim_{h
ightarrow0}\frac{\frac{12}{x + h+3}-\frac{12}{x + 3}}{h} \)

Step3: Simplify the numerator

Find a common denominator for the fractions in the numerator, which is \( (x + h + 3)(x + 3) \):

\(

$$\begin{align*} \frac{12}{x + h+3}-\frac{12}{x + 3}&=\frac{12(x + 3)-12(x + h + 3)}{(x + h + 3)(x + 3)}\\ &=\frac{12x+36-12x - 12h-36}{(x + h + 3)(x + 3)}\\ &=\frac{- 12h}{(x + h + 3)(x + 3)} \end{align*}$$

\)

Step4: Substitute back and simplify the limit

Now substitute the simplified numerator back into the derivative formula:

\( f^{\prime}(x)=\lim_{h
ightarrow0}\frac{\frac{-12h}{(x + h + 3)(x + 3)}}{h} \)

The \( h \) in the numerator and denominator (non - zero as \( h
ightarrow0 \) but \( h
eq0 \) during the limit process) cancels out:

\( f^{\prime}(x)=\lim_{h
ightarrow0}\frac{-12}{(x + h + 3)(x + 3)} \)

Step5: Evaluate the limit as \( h

ightarrow0 \)
As \( h
ightarrow0 \), we substitute \( h = 0 \) into the expression:

\( f^{\prime}(x)=\frac{-12}{(x+0 + 3)(x + 3)}=\frac{-12}{(x + 3)^2} \)

Answer:

\( f^{\prime}(x)=-\frac{12}{(x + 3)^2} \)