One of the most important applications of limits is the concept of the

*derivative of a function.* In calculus, the derivative of a function is used in a wide variety of problems, and understanding it is essential to applying it to such problems.

The **derivative** of a function *y* = *f*( *x*) at a point ( *x*, *f*( *x*)) is defined as

if this limit exists. The derivative is denoted by *f*′ ( *x*), read “ *f* prime of *x*” or “ *f* prime at *x*,” and *f* is said to be **differentiable** at *x* if this limit exists (see Figure ).

**Figure 1 **The derivative of a function as the limit of rise over run.

If a function is differentiable at *x*, then it must be continuous at *x*, but the converse is not necessarily true. That is, a function may be continuous at a point, but the derivative at that point may not exist. As an example, the function *f*( *x*) = *x* ^{1/3} is continuous over its entire domain or real numbers, but its derivative does not exist at *x* = 0.

Another example is the function *f*( *x*) = | *x* + 2|, which is also continuous over its entire domain of real numbers but is not differentiable at *x* = −2.

The relationship between continuity and differentiability can be summarized as follows: Differentiability implies continuity, but continuity *does* *not* imply differentiability.

**Example 1:** Find the derivative of *f*( *x*) = *x* ^{2} − 5 at the point (2,−1).

hence, the derivative of *f*( *x*) = *x* ^{2} − 5 at the point (2,−1) is 4.

One interpretation of the derivative of a function at a point is the **slope of the tangent line** at this point. The derivative may be thought of as the limit of the slopes of the secant lines passing through a fixed point on a curve and other points on the curve that get closer and closer to the fixed point. If this limit exists, it is defined to be the slope of the tangent line at the fixed point, ( *x*, *f*( *x*)) on the graph of *y* = *f*( *x*).

Another interpretation of the derivative is the **instantaneous velocity** of a function representing the position of a particle along a line at time *t*, where *y* = *s*( *t*). The derivative may be thought of as a limit of the average velocities between a fixed time and other times that get closer and closer to the fixed time. If this limit exists, it is defined to be the instantaneous velocity at time *t* for the function, *y* = *s*( *t*).

A third interpretation of the derivative is the **instantaneous rate of change** of a function at a point. The derivative may be thought of as the limit of the average rates of change between a fixed point and other points on the curve that get closer and closer to the fixed point. If this limit exists, it is defined to be the instantaneous rate of change at the fixed point ( *x*, *f*( *x*)) on the graph of *y* = *f*( *x*).

**Example 2:** Find the instantaneous velocity of at the time *t* = 3.

hence, the instantaneous velocity of *s*( *t*) = 1/( *t* + 2) at time *t* = 3 is −1/25. The negative velocity indicates that the particle is moving in the negative direction.

A number of different notations are used to represent the derivative of a function *y* = *f*( *x*) with *f*′ ( *x*) being most common. Some others are *y*′, *dy/dx*, *df*/ *dx*, *df*( *x*)/ *dx*, *D* _{x }*f*, and *D* _{x }*f*( *x*), and you should be able to use any of these in selected problems.