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Chapter 2: Problem 13

Find the derivative by the limit process. \(h(s)=3+\frac{2}{3} s\)

Short Answer

Expert verified
The derivative of the given function is \(\frac{2}{3}\).

Step by step solution

01

Express the function

Express the given function, which is \(h(s) = 3 + \frac{2}{3}s.\)
02

Apply the limit definition of the derivative

The limit definition of a derivative is \(h'(s) = \lim_{\Delta s \to 0} \frac{h(s+\Delta s) - h(s)}{\Delta s}\). Now you replace \(h(s+\Delta s)\) and \(h(s)\) in this formula with the given function.
03

Insert value of the function at points s and (s+螖s)

On replacing the values we get: \(h'(s) = \lim_{\Delta s \to 0} \frac{(3+\frac{2}{3}(s+\Delta s)) - (3+\frac{2}{3}s)}{\Delta s}\). After simplification, it comes out to be \(h'(s) = \lim_{\Delta s \to 0} \frac{\frac{2}{3}\Delta s}{\Delta s}\).
04

Simplify the fraction

The \(\Delta s\) in numerator and denominator cancel out each other. Thus we are left with \(h'(s) = \lim_{\Delta s \to 0} \frac{2}{3}\).
05

Determine the Limit

The term we have left doesn't depend on \(\Delta s\) at all. So the limit as \(\Delta s \to 0\) of a constant (which is \(\frac{2}{3}\)) is just that constant itself.

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