Calculus Problem of the Week September 2, 2011

Sometimes easy and sometimes hard, our calculus problem of the week could come from any calculus topic. If you really want to get better at calculus, following these problems is a great way to make yourself practice! Past calculus problems of the week.

This week’s problem was inspired by something mentioned on twitter and while it is not a difficult derivative, thinking about using the “not obvious” rule will help you look at functions in a new and different way. Try it out!
This week’s problem:
(click “see the solution” at the bottom of post to, well, see the solution.)

Find the derivative of f(x)=e^{x+5} using the product rule.

See the solution.

The “obvious” way to approach this would be with the chain rule but to make this work with the product rule, you have to go back and remember the laws of exponents.



You can rewrite e^{x+5} as (e^x)(e^5). Normally, you would be totally right in saying “well, why would you do THAT?! You just made it more complicated!” but I was trying to find a way to write it as a product in order to use the product rule. Once you have rewritten it this way:

[(e^x)(e^5)]'=(e^x)'(e^5)+(e^x)(e^5)'=(e^x)(e^5)+(e^x)(0)=(e^x)(e^5)=e^{x+5}.
(I used the fact that e^5 is a constant so its derivative is zero)

Based on this, e^{x+5} is its own derivative which makes sense since the slope of the “inside” function is 1.