How do I find and simplify $\frac{(g(t+h)-g(t))}h$ if $g(t) = 3t-5$?

Question

I start out with: $\frac{(3(t+h)-5-3t-5)}2$

simplify to:

$\frac{(3t+3h-5-3t-5)}2$

then $\frac{(3h-10)}2$

I tried entering that end it wouldn't work. I've tried entering other forms of it and it is not correct. What am I doing wrong?

Answer 1

\begin{align} \frac{g(t+h)-g(t)}{h} & = \frac{(3(t+h)-5)-(3t-5)}{h} \\ & = \frac{3t+3h-5-3t+5}{h} \\ & = \frac{3h}{h}\\ & = 3 \end{align}

I believe this is what you are after.

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Just noticed the equation is in the form of First Principles Differentiation without the limit. Since $g(x)=3t-5$, $g'(x)=3$ which is a constant term.