Understanding the Definition and Importance of the Difference Quotient in Calculating Derivatives

f(x+h)-f(x) represents the change in y in the definition of the derivative

The difference quotient, f(x+h) – f(x), represents the change in y between two points x and x+h in the function f

The difference quotient, f(x+h) – f(x), represents the change in y between two points x and x+h in the function f. In the context of the definition of the derivative, this represents the change in the dependent variable (y) when the independent variable (x) is changed by a small increment h.

The derivative of a function f(x) is the slope of the tangent line to the graph of the function at a given point. It provides the rate at which the function is changing at that specific point.

To find the derivative of a function, we use the concept of the limit. The change in y, which is represented by f(x+h) – f(x), is divided by the change in x, which is h. This gives us the average rate of change of the function between x and x+h.

As h tends to 0, the average rate of change becomes the instantaneous rate of change, also known as the derivative. Therefore, taking the limit as h approaches 0, we obtain the definition of the derivative:

f'(x) = lim(h->0) [f(x+h) – f(x)] / h

Using this definition, we can find the derivative of any function by evaluating this limit at a specific value of x.

Note that f(x+h) – f(x) represents the change in y as x changes by a small amount h. This difference in y divided by h represents the slope of the secant line passing through the points (x, f(x)) and (x+h, f(x+h)). As h approaches 0, this secant line approaches the tangent line, and the limit gives us the derivative of the function at that point.

In summary, f(x+h) – f(x) represents the change in y in the definition of the derivative, and it is a key component in finding the instantaneous rate of change of a function at a specific point.

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