To estimate the derivative \(g'(1)\) of the function \(g(t) = \frac{6}{t^5}\), we can use the definition of the derivative.
The derivative of a function at a specific point represents the rate of change of the function at that point. In this case, we want to find the rate of change of \(g(t)\) at \(t = 1\), which is denoted as \(g'(1)\).
The definition of the derivative is:
\[g'(1) = \lim_{h \to 0} \frac{g(1 + h) - g(1)}{h}\]
To estimate the value of \(g'(1)\), we need to evaluate the expression on the right-hand side of the equation. Let's substitute the values into the equation:
\[g'(1) = \lim_{h \to 0} \frac{g(1 + h) - g(1)}{h}\]
Substituting \(g(t) = \frac{6}{t^5}\) into the equation, we get:
\[g'(1) = \lim_{h \to 0} \frac{\frac{6}{(1 + h)^5} - \frac{6}{1^5}}{h}\]
Simplifying the expression further, we have:
\[g'(1) = \lim_{h \to 0} \frac{\frac{6}{(1 + h)^5} - 6}{h}\]
To compute the limit, we substitute \(h = 0\) into the expression:
\[g'(1) = \frac{\frac{6}{(1 + 0)^5} - 6}{0}\]
However, dividing by zero is undefined, and the limit does not exist. Therefore, we cannot estimate \(g'(1)\) using this method.
In conclusion, the estimated value of \(g'(1)\) is undefined using the method of evaluating the limit of the difference quotient.