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)(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)(f(3+h)-f(3))/(h) \\ f^(\prime)(3)=\lim _(h\to0)((3+h)^2-5-(3^2-5))/(h) \end{gathered}`

Operating:

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