$f'(1)$ represents the instantaneous rate of change or slope of the function $f(x)$ at the point $x=1$. It gives us the rate at which the function is changing at that specific point.
To evaluate $f'(1)$, we use the power rule for derivatives and the rule for the derivative of natural logarithms. First, we find the derivative of $f(x)$, which is $f'(x)=7+\frac{1}{x}$. Then, we substitute $x=1$ into the derivative, giving us $f'(1)=7+\frac{1}{1}=8$. Therefore, $f'(1)=8$.
The value of $f'(1)$ tells us the slope of the function at the specific point $x=1$, which can be interpreted as the rate of change of the function at that point. It also helps us determine the behavior of the function near $x=1$, such as whether it is increasing or decreasing.
Yes, $f'(1)$ is the same as the derivative of $f(x)$ at the point $x=1$. The derivative of a function gives us the slope of the function at any given point, and when we substitute a specific value for $x$, we get the derivative at that point, which is $f'(1)$ in this case.
The equation of the tangent line at $x=1$ is given by $y=f'(1)(x-1)+f(1)$. We can use the value of $f'(1)$ that we calculated earlier and the given point $(1,f(1))$ to find the equation of the tangent line. This equation represents a line that is tangent to the curve of $f(x)$ at $x=1$ and has the same slope as the curve at that point.