Question

(a) the derivative of a function f at x, denoted by f(x), is f(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} provided that the limit exists. use the definition of the derivative of f at x to find the derivative of the given function f(x) when x = 4.

f(4) = \lim_{h \to 0} \frac{\frac{1}{4 + h} - \frac{1}{x}}{h}

evaluate the limit expression to find f(4).

f(4) = \square (type an integer or a fraction.)

Explanation:

Step1: Simplify the numerator

First, simplify \(\frac{1}{4 + h}-\frac{1}{4}\).

$$
LATEXBLOCK0
$$

So, \(f^{\prime}(4)=\lim_{h
ightarrow0}\frac{\frac{-h}{4(4 + h)}}{h}\).

Step2: Simplify the fraction

Simplify \(\frac{\frac{-h}{4(4 + h)}}{h}\).

$$
LATEXBLOCK1
$$

Step3: Evaluate the limit

Now, find \(\lim_{h
ightarrow0}\frac{-1}{4(4 + h)}\).
Substitute \(h = 0\) into \(\frac{-1}{4(4 + h)}\), we get \(\frac{-1}{4\times(4+0)}=-\frac{1}{16}\).

Answer:

\(-\frac{1}{16}\)