Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the derivative of $\ln(4^x)$ (which I believe is also equal to $x\ln4$)?

Is it $\dfrac{1}{x\ln4}$?

share|improve this question
2  
as you said, $ln(4^x)$ = $xln4$, so its derivative is simply $ln4$. –  the L May 15 '13 at 7:04
    
or did you mean $\left(\ln{4} \right)^x$ ? –  DanZimm May 15 '13 at 7:05
2  
See this page for an explanation of how to write math here. –  Zev Chonoles May 15 '13 at 7:05
    
Sorry, I have now edited the post to be more clear. –  Gannicus May 15 '13 at 7:08
    
@Inceptio Thanks for the dfrac, I didn't know about that LOL –  Gannicus May 15 '13 at 7:16

4 Answers 4

up vote 2 down vote accepted

Recall that for $a>0$, we have $$\log(a^b) = b \log(a)$$ Also, note that $$\dfrac{d}{dx}\left( cx\right) = c$$ I trust you can finish it from here.

share|improve this answer
    
The concept of ln, e, log always boggles me. I keep forgetting that they are constants and not scary math figures. –  Gannicus May 15 '13 at 7:19

if your question is $ln(4^x)$ then its answer will be $\ ln4$ and if your question is $(\ln4)^x$ then use this formula $$\dfrac {d}{dx}a^x=a^x\cdot\ln{a}$$ so $$\dfrac {d}{dx}{(\ln4)}^x=(\ln4)^x\cdot\ln{(\ln4)}$$

share|improve this answer
2  
I've given answer of both possibilities of question so please check carefully who downvote it –  iostream007 May 15 '13 at 7:19
    
(+1). For providing both instances. The down-voter should definitely consider leaving a comment. –  Inceptio May 15 '13 at 7:24
    
peoples are too fast to downvote without reading whole thing and don't want to write reason –  iostream007 May 15 '13 at 7:26

Hint : $\ln 4^x=y \implies x \ln 4=y$

Since $\ln4$ is a constant. The derivative is simply the constant, i.e $\ln 4$

share|improve this answer

The thing you are supposed to understand here is that constants don't do anything to derivatives -- they just factor out of the expression. More precisely, we have for any function $f$ and a constant $c$ not depending on $x$ that $$ \frac{d}{d x} c f(x) = c \frac{d}{d x} f(x) $$ For example, since $\frac{d}{d x} (x^4) = 4x^3$, I can multiply by any constant and the derivative is just as straightforward: $$ \frac{d}{d x} \left[ 2.33\pi^2 \cos (4) \right] x^4 = \left[ 2.33\pi^2 \cos (4) \right] 4x^3 $$

This works since the $ 2.33\pi^2 \cos (4) $ is constant - it doesn't depend on $x$.

If you get this, you should be able to calculate derivatives like this easily.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.