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How to Compute Derivatives

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will briefly review what the derivative of a function is. Then we will look at the limit definition of a derivative, use it to compute derivatives, and look at a few shortcuts that result from the limit definition of derivatives.

Derivatives

Suppose you've just watched a car race on an out-and-back course. The drivers drove 2800 feet out and 2800 feet back. The winner of the race drove in such a way that her distance from the start can be modeled using the function:

  • f(x) = -7x2 + 280x, where x is the number of seconds since the start of the race.

As she was bragging about first place, someone asked her how fast she was going. She realized that she knew her speed was different at different points in the race, but she wasn't sure how to find how fast she was going at any given point.

How fast is the car going at any given point?
car race

Hmmm…any ideas? Thankfully, there is a mathematical answer to this conundrum, and that answer lies in derivatives.

The derivative of a function is the rate at which the function value is changing, with respect to x, at a given value of x. Therefore, if we can find the derivative of the winner's distance function, then we can find how fast she was going at any given time in the race. Let's take a look at how to do this!

Computing Derivatives

You may recall something called the difference quotient from an algebra or pre-calculus course. The difference quotient of a function f(x) is a formula that gives the slope of the line through any two points with x-coordinates x and x + h on the function:

  • (f(x + h) - f(x)) / h

This is the key to computing derivatives! Derivatives are computed by finding the limit of the difference quotient of a function as h approaches 0.


compder2


Basically, we can compute the derivative of f(x) using the limit definition of derivatives with the following steps:

  1. Find f(x + h).
  2. Plug f(x + h), f(x), and h into the limit definition of a derivative.
  3. Simplify the difference quotient.
  4. Take the limit, as h approaches 0, of the simplified difference quotient.

Example

So, consider our racing function f(x) = -7x2 + 280x. First, we find f(x + h):

  • f(x + h) = -7(x + h)2 + 280(x + h) = -7(x2 + 2xh + h2) + 280x + 280h = -7x2 - 14xh - 7h2 + 280x + 280h

Now, we plug into the limit definition, simplify, and find the limit.


compder3


We see that the derivative of f(x) is

  • f ' (x) = -14x + 280

We can use this formula to calculate the winner's speed at any time during the race. For instance, consider her speed after 10 seconds. We plug x = 10 into the derivative formula:

  • f ' (x) = -14(10) + 280 = 140

We get that the derivative of f at x = 10 is 140, so at 10 seconds into the race, she was driving at 140 mph - wow! That's so fast!

Another Example

Okay, one more example of using this limit definition to compute a derivative. Consider the function g(x) = 1 / x, where x ≠ 0. To find the derivative using the limit definition of derivatives, we first find g(x + h).

  • g(x + h) = 1 / (x + h)

Now, we plug g(x + h), g(x), and h into the limit definition and find the limit.


compder4


We see that the derivative of g(x) = 1 / x is g ' (x) = -1 / x2.

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