Simplifying Derivatives

[tfs985]

Hi All:

I have 3 derivatives problems that I need to simplify in order to find relative extrema. As crazy as this sounds, I am comfortable with calculus, but I have major issues in the Algebra department. Can someone check out these 3 problems and let me know what to do, because I am completely lost!

1) f(x) = (3x-8)^7/5

Using the general powers rule, I got a derivative of: 7/5 (3x-8)^2/5 (3). How can I simplify this further?

2) h(x) = x^15e^-3x

Using the product rule, I got a derivative of: (x^15) (-3e^-3x) + (e^-3x) (15x^14). Again, how should I go about simplifying this?

3) f(x) = x/x^2 + 58

Using the quotient rule, I got a derivative of: (x^2 + 58) (1) - (x) (2x) / (x^2 + 58)^2. I'm pretty sure this one has no solution, but if anyone can verify this I would be elated.

Thanks in advance for the help!

 

[srmichael]

For the first one, about all you can do is multiply \(\displaystyle \frac{7}{5}\times3\)

For the second one, do you see you can factor out a \(\displaystyle 3x^{14}e^{-3x}\)?

For the third one, on the contrary. There is a solution. Combine like terms in the numerator and what do you get?

 

[lookagain]

1) f(x) = (3x-8)^(7/5)

Using the general powers rule, I got a derivative of: (7/5)(3x-8)^(2/5)(3). \(\displaystyle \text{You \ could \ get \ by \ without \ grouping \ symbols \ around \ the \ 7/5, }\)

\(\displaystyle \text{ but \ I \ used \ them \ for\ greater \ consistency.}\)



2) h(x) = x^15e^(-3x)

Using the product rule, I got a derivative of: (x^15)[-3e^(-3x)] + [e^(-3x)](15x^14).



3) f(x) = x/(x^2 + 58)

Using the quotient rule, I got a derivative of: [(x^2 + 58)(1) - (x)(2x)]/(x^2 + 58)^2.

\(\displaystyle tsf985,\)

\(\displaystyle \text{you must use grouping symbols as in the above because of the }\) \(\displaystyle \text{Order of Operations.}\)


Even additional grouping symbols than I used might be added to give even
more clarification and/or emphasis.


Amended example: (7/5)[(3x - 8)^(2/5)](3)

 

[summer.johnson]

can someone help me now? as in now urgent! simplifying derivatives

(v^2)(6/√12^v-1)+(√12^v-1)(2v)