Question

Use the limit definition to find the derivative of the function. $$ g(s)=\frac{1}{3} s+2 $$

Short answer

The derivative of the function is \(\frac{1}{3}\).

Step-by-step solution

Define the function

Firstly, identify the function which is \(g(s)=\frac{1}{3} s+2\).

Apply the limit definition of derivative

Use the definition of the derivative which is the limit as h approaches 0 of (f(x+h) - f(x))/h. So in this case, (g(s+h) - g(s))/h is used.

Calculate g(s + h)

Calculate g(s + h) by replacing every s in the function with (s + h) which gives \(\frac{1}{3} (s + h) + 2\).

Substitute into the derivative definition

Substitute g(s + h) and g(s) into the derivative definition to get \([ \frac{1}{3} (s + h) + 2 - ( \frac{1}{3} s + 2 ) ] / h\). Simplify this to get \(\frac{1}{3}\).

Calculate the limit

As this doesn't depend on h anymore, the limit as h approaches 0 is simply \(\frac{1}{3}\).