Weekly Math Problem #65

Definition of a Derivative. One of the nice things about math is that rules just work. Without knowing why rules work, we are able to use them for our purposes. 😁 Derivatives are a classic example of this! I'm so glad that I can just use derivative rules to find the derivatives of functions, instead of using the definition of a derivative all the time. I remember learning precalculus/calculus and being introduced to and having to use the definition of the derivative. Once I learned derivative rules, I breathed a sigh of relief. The definition required lots of writing and, sometimes, fancy algebra tricks. 🙄

Despite my laziness, we're going to use the limit definition of the derivative this week. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to use a formula (substitution)
        ✔️     How to find a limit
        ✔️     Simplifying fractions

     

WMP! #65 says...

Happy solving!

Check back on Saturday, November 27th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Let's get into the solution for this week's problem! 

The solution requires us to use substitution with the limit definition of the derivative. Then, evaluate the limit.

Houston...we have a problem! When evaluating the limit, we end up with an undefined, rational expression. 🤔 This doesn't tell me anything about the derivative of the function. What is does tell me is that another technique is needed to evaluate the limit. 

What we're going to do is rationalize the numerator. How do I know that? ...From experience. Remembering this technique comes from repetition over the years.  Rationalizing the numerator will yield an equivalent expression that allows us to evaluate the limit.

Now, we can replace the equivalent expression and evaluate the limit.

Now we have a function for the first derivative of the square root of x. 👍🏿 

There is a quicker way to get to this result...and that's by using the power rule for derivatives. 

I don't know about you, but I am grateful for rules and shortcuts. 😁

▪️ Were you able to find the first derivative using the limit definition??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for sticking around and solving with me this week
...I truly appreciate it!
Of course, WMP! #66 is up next. 💪🏿

Cheers!

The Younge Lady