Question

I'm trying to evaluate a derivative. yikes!
its the derivative of h with respect to t of
h = 80-5t^2

I'm using the difference quotient and initially tried:
h(4)-h(0)/4 but this is wrong, it gave me -20 (the right answer is -40).
I next tried (h(4+dt)-h(4))/dt where dt is delta time as it approaches 0... now Im stuck.

Answer 1

the correct way to evaluate this is as follows:\[\begin{align}
\frac{dh}{dt}&=\lim_{dt\rightarrow 0}(\frac{h(t+dt)-h(t)}{dt})\\
&=\lim_{dt\rightarrow 0}(\frac{80-5(t+dt)^2-(80-5t^2))}{dt})\\
&=\lim_{dt\rightarrow 0}(\frac{80-5(t^2+2tdt+(dt)^2)-80+5t^2}{dt})\\
&=\lim_{dt\rightarrow 0}(\frac{80-5t^2-10tdt-5(dt)^2-80+5t^2}{dt})\\
&=\lim_{dt\rightarrow 0}(\frac{-5dt(2t-dt)}{dt})\\
&=\lim_{dt\rightarrow 0}(-5(2t-dt))\\
&=-10t
\end{align}\]then just evaluate this at the given value of t.

Answer 2

we could just use the power rule\[na ^{n-1}\]