C.4 Chain Rule - Used to Find Yet More Derivatives

On this screen, we're going to use the Chain Rule to find yet more derivatives, including 𝑎𝑥, sec⁡𝑥, cot⁡𝑥, and more. This will both (1) give you even more crucial practice at using the Chain Rule, and (2) will let you find the derivative of whatever functions you encounter, plus learn a cool fact about even and odd functions.

Derivative of 𝑎𝑥

Let's see first how we can easily extend our new knowledge that 𝑑𝑑𝑥𝑒𝑢 =𝑒𝑢𝑑𝑢𝑑𝑥 to find the derivative of 𝑎𝑥, now much more easily than we could before we had the Chain Rule as a tool.

Practice Problem #1: Derivative of 𝑎𝑥

Show that 𝑑𝑑𝑥𝑎𝑥 =𝑎𝑥ln⁡𝑎, where 𝑎 is a constant and 𝑎 >0. (For example, 𝑑𝑑𝑥2𝑥 =2𝑥ln⁡2.)

Hint: Start with 𝑎𝑥 =𝑒ln⁡𝑎𝑥, and use 𝑑𝑑𝑥𝑒𝑢 =𝑒𝑢 ⋅𝑑𝑢𝑑𝑥.

By the way, notice that the result of Problem #1, 𝑑𝑑𝑥𝑎𝑥 =𝑎𝑥ln⁡𝑎, still applies when 𝑎 =𝑒 : 𝑑𝑑𝑥𝑒𝑥=𝑒𝑥ln⁡(𝑒)1=𝑒𝑥

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You probably have firmly in mind that 𝑑𝑑𝑥𝑒𝑥 =𝑒𝑥, and so just need to remember that if the base of the exponential a is anything other than e, you have to multiply the derivative by ln⁡𝑎. For instance, 𝑑𝑑𝑥3𝑥 =3𝑥ln⁡3.

Quotient Rule from Chain and Product Rules

Let's next use the Chain Rule as an alternate way to develop the Quotient Rule, starting from the Product Rule.

Practice Problem #2: Quotient Rule from Chain and Product Rules
Use the Chain Rule and the Product Rule to develop the Quotient Rule. Start from 𝑓(𝑥)𝑔(𝑥) =𝑓(𝑥)(𝑔(𝑥))−1.

Derivatives of sec⁡𝑥 and cot⁡𝑥

And let's find some more common trig derivatives.

Derivatives of Even and Odd Functions

The next problem develops a slick insight into the derivatives of even and odd functions.

Practice Problem #4: Derivatives of even and odd functions

Show the following.

(a)
Prove that the derivative of an even function is an odd function. [Recall that for an even function, 𝑓( −𝑥) =𝑓(𝑥).]
(b)
Prove that the derivative of an odd function is an even function. [Recall that for an odd function, 𝑓( −𝑥) = −𝑓(𝑥).]

This screen concludes our focus on the Chain Rule, though of course we'll be using it often as we proceed.

For now, what are your thoughts about using this crucial rule? What advice would you give to someone just starting to learn what it's about and how to use it? What remaining questions do you have? Please post on the Forum and let the Community know!