How to Find the Derivative of a Function Using the Difference Quotient

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

To find the derivative of a function f(x) at a point x=c, you need to evaluate the following limit:

f ‘(c) = lim(x->c) [f(x) – f(c)] / (x – c)

This is known as the difference quotient, and it represents the rate of change of f(x) at x=c

To find the derivative of a function f(x) at a point x=c, you need to evaluate the following limit:

f ‘(c) = lim(x->c) [f(x) – f(c)] / (x – c)

This is known as the difference quotient, and it represents the rate of change of f(x) at x=c.

To calculate the limit, you need to simplify the expression inside the limit as much as possible and then substitute the value x=c. Let’s go through an example to make it clearer.

Example:
Suppose we have the function f(x) = 2x^2 + 3x – 1, and we want to find f ‘(c) at x=c.

Step 1: Start with the difference quotient expression:
lim(x->c) [f(x) – f(c)] / (x – c)

Step 2: Substitute the function f(x) into the difference quotient:
lim(x->c) [(2x^2 + 3x – 1) – (2c^2 + 3c – 1)] / (x – c)

Step 3: Simplify the numerator:
lim(x->c) [2x^2 + 3x – 1 – 2c^2 – 3c + 1] / (x – c)

Step 4: Combine like terms in the numerator:
lim(x->c) [2x^2 – 2c^2 + 3x – 3c] / (x – c)

Step 5: Now, you can cancel out common factors between the numerator and denominator if they exist. In this case, there are no common factors, so we cannot further simplify.

Step 6: Finally, substitute x=c into the expression:
lim(x->c) [2c^2 – 2c^2 + 3c – 3c] / (c – c)

Step 7: Simplify the numerator:
lim(x->c) (0) / (0)

Step 8: At this point, you might think the limit is undefined because you have 0/0. However, you should not conclude the derivative is undefined from this expression alone. Instead, you need to find alternate methods such as using the quotient rule or common factor cancellation to find the derivative. In this case, the derivative of f(x) will be 0.

The above example showcases the process of finding the derivative using the difference quotient. It’s important to note that not all functions can be easily differentiated using this method, especially more complex functions. In such cases, other techniques like the power rule, product rule, chain rule, quotient rule, or a combination of these rules are applied.

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