The Ease of Derivatives

Suppose you are worried that your speedometer is incorrectly calibrated. So you drive at a constant speed and count mile markers on the freeway. After three hours you count 180 mile markers. How fast are you going? This is an easy problem because your speed is constant, but it can easily get more difficult. Once you start adding in acceleration it gets more difficult. And that is where derivatives help.

Integrals tell us how far we have travelled. Derivatives tell us how fast we are travelling. If you look at a graph, the derivative corresponds to the rate of change of the function. It is a measure for how steep the graph is at a given point.

Finding the Derivative

Lets start with y = 3x. You probably already know the rate of change, or slope, of this graph. It is 3. But you could easily find out by picking two arbitrary points and calculating the difference in values of the function by the values of the points themselves. Lets pick 1 and 5 as our two points.

slope = (3×5 – 3×1) ÷ (5 – 1) = 12 ÷ 4 = 3

We run into an immediate problem if we try this on the graph of a parabola such as y = x². The rate of change is different depending on which points you choose. However, there is an solution. Just pick to points that are very close together. We want to find the limit at point x, so lets make the other point x + h, with h being the distance between the two points.

slope = [ (x + h)² - x² ] ÷ h
= [ x² + 2hx + h² - x² ] ÷ h
= [ 2hx + h² ] ÷ h
= 2x + h
= 2x

Since we want the two points to be as close together as possible, we let h = 0 and the answer is 2x. (Yes, this blithely ignores all the rigor of limits and continuity, but the goal is to understand the concept, not to get bogged down with limits and epsilon-deltas. But rest assured that it does all work out).

The Power Rule

Now lets try y = x³

slope = [ (x + h)³ - x³ ] ÷ h
= [ x³ + 3hx² + 3xh³ + h³ - x³ ] ÷ h
= [ 3hx² + 3xh² + h³] ÷ h
= 3x² + 3xh + h²
= 3x²

Once again, we are relying on letting h = 0 while ignoring the issue of limits and continuity.

Now lets try y = x⁴

slope = [ (x + h)⁴ - x⁴ ] ÷ h
= [ x⁴ + 4hx³ + 6x²h² 4xh³ + h⁴ - x⁴ ] ÷ h
= [ 4hx³ + 6x²h² + 3xh³ + h⁴] ÷ h
= 4x³ + 6x²h + 4xh² + h³
= 4x³

Do you see the pattern? Here are what we have found. If the slop

slope( x² ) = 2x
slope( x³ ) = 3x²
slope( x⁴ ) = 4x³

Generally, the slope of xⁿ = nx^n-1. This is called the power rule. This can be proven formally using mathematical induction.

Lets find the derivative for 7x².

slope = [ 7(x + h)² - 7x² ] ÷ h
= 7[ x² + 2hx + h² - x² ] ÷ h
= 7[ 2hx + h² ] ÷ h
= 7 [2x + h]
= 14x

This leads to a more general version of the power rule.

slope( axⁿn) = n·ax^n-1

This more general version should technically be proven using mathematical induction, but hopefully you see from how we were immediately able to factor the 7 in the previous example that the pattern should hold.

Exercises

1. Calculate the derivate of 14x⁹
2. Calculate the derivate of 4x²⁹
3. Go through the method of finding a derivative for y = 3x³. Then compare your answer to the power rule.