Question

Find the derivative of the function at the given numberf(x)=x^2 - 5 when x=3 using the difference quotient f(x+h) - f(x) over h

Answer 1

Derivative

Given a function f(x), the derivative of f(x) in a point x = a can be found by using the definition:

$f^{\prime}(a)=\lim _{h\to0}\frac{f(a+h)-f(a)}{h}$

Given:

$\begin{gathered} f(x)=x^2-5 \\ a=3 \end{gathered}$$\begin{gathered} f^{\prime}(3)=\lim _{h\to0}\frac{f(3+h)-f(3)}{h} \\ f^{\prime}(3)=\lim _{h\to0}\frac{(3+h)^2-5-(3^2-5)}{h} \end{gathered}$

Operating:

$\begin{gathered} f^{\prime}(3)=\lim _{h\to0}\frac{9+6h+h^2-5-4}{h} \\ f^{\prime}(3)=\lim _{h\to0}\frac{6h+h^2}{h} \\ \text{Factoring:} \\ f^{\prime}(3)=\lim _{h\to0}\frac{h(6+h)}{h} \\ \text{Simplifying:} \\ f^{\prime}(3)=6 \end{gathered}$