Estimate Derivatives Given Function Values

What is the value of $f(x)$ at $x=0$?

Which of the following denotes the derivative of $f(x)$ at $x=0$?

Since we cannot determine the limit explicitly, what information would be helpful for estimating the limit?

Great! Let's take $h=1$. What estimate for the derivative of $f(x)$ at $x=0$ does this give us?

The actual value of the derivative is $1$. How could we get a better estimate of the derivative?

Consider the function $f(x) = x^2$. We wish to find the derivative of $f(x)$ at $x = 1$. Which of the following is equal to $f'(1)$?

Using the value $h = 1$, what estimate do you get for $f'(1)$?

Using the value $h = 0.5$, what estimate do you get for $f'(1)$?

Using the value $h = 0.1$, what estimate do you get for $f'(1)$?

Using the value $h = 0.01$, what estimate do you get for $f'(1)$?

By now, you might be starting to see a pattern. For an arbitrary nonzero value of $h$, what estimate do you get for $f'(1)$?

As $h$ approaches $0$, what value does the estimate approach?

What is $f'(1)$?

Using a similar logic, can you find $f'(x)$ for any value of $x$?