The basic rules of differentiation are linearity, the product rule, and the chain rule. Once we start graphing functions, we’ll revisit these rules.
The linearity of differentials means
and are constants, while and might change.
This looks obvious, but here’s a quick sketch.
First we’ll look at . Construct a right triangle with base 1 and hypotenuse . Then extend the base by length . This creates a larger, similar triangle. The hypotenuse must be times the base, so the hypotenuse is extended by .
Then increase by . This induces an increase in the hypotenuse.
The little right triangle made by and is similar to the original, so
Next look at . is just two line segments laid one after the other. We increase the lengths by and and see what the change in the total length is.
These rules combine to give the rule for linearity
The Product Rule
The product rule is
To show this, we need a line segment with length .
Start by drawing , then drawing a segment of length 1 starting at the same place as and going an arbitrary direction.
Close the triangle. Extend the segment of length 1 by , and close the new triangle. We’ve now extended the base by .
Increase by and by . This results in several changes to .
The segment has a little bit chopped off on the left, since cuts into the place where it used to be.
is also extended twice on the right. The first extension is the projection of down onto the base. All such projections multiply the length by , so the piece added is .
Finally there is a piece added from the very skinny tall triangle. It is similar to the skinny, short triangle created by adding to . The tall triangle is times as far from the bottom left corner as the short one, so it is times as big. Since the base of the short one is , the base of the tall one is .
Combining all three changes to , one subtracting from the left and two adding to the right, we get
This is the product rule. We’ll give another visual proof in the exercises.
The Chain Rule
Suppose we want . (There’s no particular reason I can think of to want that, but we have a limited milieu of functions at hand right now.)
We know . Let .
But we already know that , so substitute that in to get
This is called the chain rule. A symbolic way to right it is
Suppose you are hiking up a mountain trail. is your height above sea level. is the distance you’ve gone down the trail. is the time you’ve been hiking.
is the rate you are gaining height. According to the chain rule, you can calculate this rate by multiplying the slope of the trail to your speed .
- Show that the linearity rule is a special case of the product rule.
- What is the derivative of with respect to ? Take the derivative with respect to of that. (This is called a “second derivative”.) What do you get? (Answer: -1 times the original function)
- Use the product rule to prove by induction that the derivative of is for all positive integers .
- Apply the product rule to to prove that the “power rule” from the previous question holds for all integers .
- Look back at the arguments from the introduction. Draw a rectangle with one side length and one side length . Its area is . Use this to prove the product rule.
- Apply the chain rule to to find the derivative of with respect to for all integers (Answer: )
- Argue that the derivative of for all rational numbers .
- Show that the derivative of a polynomial is always another polynomial. Is there any polynomial that is its own derivative? (Answer: no, except zero)
- Combine the product rule with the chain rule to prove the quotient rule