Question

(a) the derivative of a function f at x, denoted by f(x), is f(x) = limₕ→0 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) = limₕ→0 (1/(4 + h)) - (1/x) / h
evaluate the limit expression to find f(4).
f(4) = -1/16 (type an integer or a fraction.)
(b) f(4) = 1/16 (type an integer or a fraction.)

Explanation:

Step1: Simplify the numerator

$$\begin{align*} \frac{1}{4 + h}-\frac{1}{4}&=\frac{4-(4 + h)}{4(4 + h)}\\ &=\frac{4-4 - h}{4(4 + h)}\\ &=\frac{-h}{4(4 + h)} \end{align*}$$

Step2: Substitute into the limit expression

$$\begin{align*} f^{\prime}(4)&=\lim_{h ightarrow0}\frac{\frac{-h}{4(4 + h)}}{h}\\ &=\lim_{h ightarrow0}\frac{-h}{4(4 + h)\cdot h}\\ &=\lim_{h ightarrow0}\frac{-1}{4(4 + h)} \end{align*}$$

Step3: Evaluate the limit

Substitute \(h = 0\) into \(\frac{-1}{4(4 + h)}\)
\(\frac{-1}{4(4+0)}=-\frac{1}{16}\)

Answer:

\(f^{\prime}(4)=-\frac{1}{16}\)