Differentiating natural logs and exponential functions?

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SUMMARY

This discussion focuses on differentiating the functions \(\frac{e^{3^x}}{\ln x}\) and \(\ln(e^{-2^x} + e^{-x})\). The participants clarify that the derivative of \(\ln x\) can be computed, and emphasize the importance of using the quotient rule and product rule for differentiation. The correct approach involves recognizing that \(\frac{d}{dx} \frac{f(x)}{g(x)}\) does not equal \(\frac{f'(x)}{g'(x)}\), and participants suggest researching these differentiation rules for better understanding.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically differentiation.
  • Familiarity with the quotient rule for differentiation.
  • Knowledge of the product rule for differentiation.
  • Ability to differentiate logarithmic functions, specifically \(\ln x\).
NEXT STEPS
  • Research the quotient rule in calculus for differentiating functions in the form \(\frac{f(x)}{g(x)}\).
  • Study the product rule for differentiating products of functions.
  • Practice differentiating logarithmic functions, including \(\ln x\) and its derivatives.
  • Explore examples of differentiating exponential functions, particularly those involving constants and variables.
USEFUL FOR

Students studying calculus, educators teaching differentiation techniques, and anyone looking to improve their understanding of logarithmic and exponential function differentiation.

Mirth
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Hey all, I'm really having a hard time figuring out a couple of problems in which I have to differentiate:

1: \frac{e^3^x}{\ln x}

I just don't know how to put it together... I know that e^3^x is 3e^3^x, and I know that you can't different \ln x, so I don't know what to do from there...

And:

2: \ln(e^-^2^x + e^-^x)

Totally not sure what to do there... Any help would be appreciated, I've looked online for similar examples and couldn't really find anything relevant to help me. Thanks!
 
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Let's take the first question first,
Mirth said:
Hey all, I'm really having a hard time figuring out a couple of problems in which I have to differentiate:

1: \frac{e^3^x}{\ln x}

I just don't know how to put it together... I know that e^3^x is 3e^3^x
Correct.
Mirth said:
and I know that you can't different \ln x, so I don't know what to do from there...
Erm... yes you can!

HINT: Let y = lnx, then x = ey. Now differentiate x = ey with respect to x.
 
Hootenanny said:
Let's take the first question first,

Correct.

Erm... yes you can!

HINT: Let y = lnx, then x = ey. Now differentiate x = ey with respect to x.

Hrm, I'm feeling a bit retarded so bear with me, hehe. So, \frac {1}{x}?
 
Mirth said:
Hrm, I'm feeling a bit retarded so bear with me, hehe. So, \frac {1}{x}?
Correct :biggrin:

So as far as putting it all together, can you suggest anything? Perhaps some sort of rule...?
 
I'm really retarded at math, ugh... Hehe. So I have \frac {3e^3^x}{1/x} ... Pretty sure I'm lost on what to do from there. :(
 
Mirth said:
I'm really retarded at math, ugh... Hehe. So I have \frac {3e^3^x}{1/x} ... Pretty sure I'm lost on what to do from there. :(
No that is not correct.

\frac{d}{dx} \frac{f\left(x\right)}{g\left(x\right)} \neq \frac{f^\prime\left(x\right)}{g^\prime\left(x\right)}

What 'rules' of differentiation do you know?
 
Not sure of any rules by name... I'm going to have to research more examples, I'm so rusty in math that I am unsure...
 
Mirth said:
Not sure of any rules by name... I'm going to have to research more examples, I'm so rusty in math that I am unsure...
Try searching for the quotient rule, or the product rule and chain rule.
 
I kinda give up on the first one, heh... Can someone tell me if I have #2 right?:

Answer: \frac {1 - e^-^2^x}{e^-^x + e^-^2^x}
 
  • #10
Mirth said:
I kinda give up on the first one, heh... Can someone tell me if I have #2 right?:

Answer: \frac {1 - e^-^2^x}{e^-^x + e^-^2^x}

The quotient rule and product rule explains how to differentiate two functions that divide are divided by or multiplied by each other. Just look that up and identify f(x) and g(x) (that is the standard notation at least).

For #2, it isn't correct. What did you do to come up with that answer?
 
  • #11
Mirth, can you show us step by step how you differentiated that expression?
 
  • #12
Hi Mirth! :smile:
Mirth said:
Not sure of any rules by name...

ok, do you recognise this one by face :wink: …

(fg)' = … ? :smile:
 

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