The Derivative, Part 5

So last time, I provided a table of derivatives given a function that is of a particular form. Because of rules 3 and 4 (you will need to see my last post to see what these are), along with the other entries in the table, you can now differentiate many functions not specifically in the table. But there are still many functions that you cannot differentiate without other rules. For example, if

\[f( x) =x^{2}\text{sin}( x)\]

there is no table entry to help you. Even though you can differentiate x² and sin(x) separately, there is no rule in the table that allows you to differentiate their multiplication together since they are both functions of x, that is, neither one is just a constant. You can’t use rule 4 here.

There is a differentiation rule that handles this. It is the multiplication rule and it states that if you have a function of the form

\[f( x) =u( x) v( x)\]

then the derivative is

\[f'( x) =u( x) v'( x) +u'( x) v( x)\]

This can be proven using the basic definition of a derivative, but you can just take my word for it.

So in the example at the beginning of this post,

u( x) =x^{2} ,\ \ \ \ \ v( x) =\text{sin}( x)\\
u'( x) =2x,\ \ \ \ v'( x) =\text{cos}( x)

where I used the table in my last post to find the individual derivatives. So according to the rule,

\[f'( x) =x^{2}\text{cos}( x) +2x\text{sin}( x)\]

Now this rule can be extended to handle more than two functions multiplied together. If

\[f( x) =u( x) v( x) w( x)\]

then you can use the original rule twice, or

\[ f'( x) =u( x) v( x) w'( x) +u( x) v'( x) w( x) +u'( x) v( x) w( x)\]

I think you can see the pattern here. So if

f( x) =x^{2}\text{sin}( x)\text{cos}( x)\\
u( x) =x^{2} ,\ \ v( x) =\text{sin}( x) ,\ \ \ w( x) =\text{cos}( x)\\
u'( x) =2x,\ \ v'( x) =\text{cos}( x) ,\ \ w'( x) =-\text{sin}( x)

So the derivative is

\[f'( x) =-x^{2}\text{sin}^{2}( x) +x^{2}\text{cos}^{2}( x) +2x\text{sin}( x)\text{cos}( x)\]

Now this can be simplified using trig identities but I will leave it here.

What about a function that’s a division of two functions? Yes there is rule for that as well, but I’ll cover that in my next post.