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We give explanation for the product rule and chain rule.
Now that we know about differentials, let’s use them to give some intuition as to why
the product and chain rules are true.
Explanation of the product rule
Linear approximations can help us explain why the product rule works.
The product rule If and are differentiable, then
To start, we need some way to
understand the function One interpretation of multiplication is it is the area of a
To understand the derivative of the product, we must understand how the area, ,
changes as changes. If we change the inputs of and by , then the size of the
However, we know from our previous work that
so now our picture becomes:
Note, if we think of , then we can also label our picture as follows:
Finally, from the pictures above and recalling that
we see that
Dividing both sides by we see and letting go to zero we see
Explanation of the chain rule
Now we’ll use linear approximations to help explain why the chain rule is
Chain Rule If and are differentiable, then
We’ll try to understand this
geometrically. In what follows, the functions and look like lines; however, the young
mathematician should realize that we are not looking at true lines, instead we are
looking at and sufficiently “zoomed-in” so that they appear to be lines. First
consider a graph of with respect to :
Now consider a graph of with respect to :
If we combine these graphs, by laying the graph of on its side, we obtain:
Ah! From this we see that
These “explanations” are not meant to be the end of the story for the product rule
and chain rule, rather they are hopefully the beginning. As you learn more
mathematics, these explanations will be refined and made precise.