## f ‘(x) is the limit of the following difference quotient as x approaches c

### To find the derivative, f'(x), we need to find the limit of the difference quotient as x approaches c

To find the derivative, f'(x), we need to find the limit of the difference quotient as x approaches c. The difference quotient is a way to approximate the derivative of a function at a specific point.

The difference quotient formula is given by:

f'(c) = lim (h -> 0) [f(c + h) – f(c)] / h

Here, c is the point at which we want to find the derivative and h is a small increment or change in x.

To use the difference quotient formula, we need the function f(x) and the specific point c.

Let’s say we have a function f(x) = x^2 and we are looking for f'(3), which means finding the derivative of f(x) at x = 3.

Now, substitute the values in the difference quotient formula:

f'(3) = lim (h -> 0) [f(3 + h) – f(3)] / h

Substitute f(x) = x^2:

f'(3) = lim (h -> 0) [(3 + h)^2 – 3^2] / h

Expand (3+h)^2:

f'(3) = lim (h -> 0) [9 + 6h + h^2 – 9] / h

Simplify by canceling out the 9s:

f'(3) = lim (h -> 0) [6h + h^2] / h

Now simplify the expression by factoring out an h:

f'(3) = lim (h -> 0) h(6 + h) / h

Cancel out h:

f'(3) = lim (h -> 0) (6 + h)

Now take the limit as h approaches 0:

f'(3) = 6 + 0

f'(3) = 6

Therefore, the derivative of the function f(x) = x^2 at x = 3 is equal to 6.

Note: This is just an example to illustrate the process of finding the derivative using the difference quotient. The actual function and point of evaluation may vary.

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