## the definition

Here, we give the formal definition of the derivative:

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

or, the value of the derivative of a function at a certain point is equal to the slope of the tangent line at that point.

it's really this simple. for quicker ways to calculate derivatives, please consult the power rule and the chain rule.

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

or, the value of the derivative of a function at a certain point is equal to the slope of the tangent line at that point.

it's really this simple. for quicker ways to calculate derivatives, please consult the power rule and the chain rule.

## The Important Limit

Here, we define the limit rigorously. First, we shall ask ourselves what great men asked themselves 300 years ago. As we can see, we are horribly late. We can calculate the average speed of an object. If a car travelled 132 miles in 2 hour, its average speed is 66 miles per hour. By the way, the interstate speed limit is 65, for those of you who have a car. The average speed. We hate to say that.

What if we wish to know the speed at a certain point of time? That would be the number on the speedometer at that point of time. Now, we define the “that point of time” to be a time period of length zero. What does that mean?

On one hand, it means a car cannot move an inch in a “point of time”. For distance needs time to accumulate. And those familiar with Red Alert should immediately recognize it as a trans-chronicle vehicle. On the other hand—the car is certain to have a velocity, or the speedometer is malfunctioning. How to find this instantaneous velocity? Though it sounds incredible, we divide distance by time. (Which are both zero)

Here, we witness the first marvel of calculus—making something out of nothing. Now suppose the car is constantly moving forward at 66 miles per hour. We first chose this to be the case so that we have a strong feeling about what is going to be the answer.

If its speed is always 66 miles per hour, it’s speed at “a point of time” is obviously 66 miles per hour. Now we solve it mathematically, using limits. We choose the function f(t), with time as the independent variable, measured in hours. We at once have y=66t, while y is in miles. Now suppose we plot this graph in the coordinate plane. It will be a straight line, passing through the origin. Obviously, the speed is the slope at that point.

The slope is given by [f(x+h)-f(x)]/h. Suppose we set h to 1. We calculate the slope as 65. Now what if we set h to 0? That would represent a time period with 0 length. In other words, it will represent a “point of time”. This operation is obviously illegal with regular arithmetic—the denominator cannot be 0. Yet, we are now armed with the weapon of limits—what is the limit of [f(x+h)-f(x)]/h at x=0? The answer would be the “instantaneous slope” at that point, or the tangent line to the graph at that point. That would be our instantaneous velocity.

evaluating the limit immediately gives us that the instantaneous velocity is 66. (use the 0/0 type strategy) as we expected.

What if we wish to know the speed at a certain point of time? That would be the number on the speedometer at that point of time. Now, we define the “that point of time” to be a time period of length zero. What does that mean?

On one hand, it means a car cannot move an inch in a “point of time”. For distance needs time to accumulate. And those familiar with Red Alert should immediately recognize it as a trans-chronicle vehicle. On the other hand—the car is certain to have a velocity, or the speedometer is malfunctioning. How to find this instantaneous velocity? Though it sounds incredible, we divide distance by time. (Which are both zero)

Here, we witness the first marvel of calculus—making something out of nothing. Now suppose the car is constantly moving forward at 66 miles per hour. We first chose this to be the case so that we have a strong feeling about what is going to be the answer.

If its speed is always 66 miles per hour, it’s speed at “a point of time” is obviously 66 miles per hour. Now we solve it mathematically, using limits. We choose the function f(t), with time as the independent variable, measured in hours. We at once have y=66t, while y is in miles. Now suppose we plot this graph in the coordinate plane. It will be a straight line, passing through the origin. Obviously, the speed is the slope at that point.

The slope is given by [f(x+h)-f(x)]/h. Suppose we set h to 1. We calculate the slope as 65. Now what if we set h to 0? That would represent a time period with 0 length. In other words, it will represent a “point of time”. This operation is obviously illegal with regular arithmetic—the denominator cannot be 0. Yet, we are now armed with the weapon of limits—what is the limit of [f(x+h)-f(x)]/h at x=0? The answer would be the “instantaneous slope” at that point, or the tangent line to the graph at that point. That would be our instantaneous velocity.

evaluating the limit immediately gives us that the instantaneous velocity is 66. (use the 0/0 type strategy) as we expected.