## anonymous one year ago Given c is some constant Why is the derivative for d/dx Log [c x] = 1/x ? And not c Log [ c x] ?

1. anonymous

I mean what kind of logical thought process should I have to lock that in ?

2. Australopithecus

Assuming it is log based 10 $\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{\ln(a)*u(x)}$

3. anonymous

I get that d/dx Log [x] = 1/x and normally c = 0 but then if I think chain rule .. I cook up .. Log [c x] (c)(x) (c) by constant multiple c Log[ c x ] But when I put Log [c x] into mathematica, it spits out 1/x ???

4. Australopithecus

its ln(x)?

5. anonymous

oh assuming oh assuming base E

6. Australopithecus

this formula still works for ln(x) because ln(e) = 1

7. phi
8. Australopithecus

The best way two approach derivatives is to split a function into smaller functions. For example for a function f(x) = ln(ax^2) notice f(x) is just two functions g(x) = ln(x) m(x) =ax^2 differentiate m(x) an g(x), we get g(x) = ln(x) g'(x) = 1/x m(x) =ax^2 m'(x) = 2ax now plug them into the chain rule formula f'(x) = g'(m(x))*m'(x) so pluging it in f'(x) = (1/(ax^2))(2ax)

9. Australopithecus

the constant cancels out

10. Australopithecus

For the derivative

11. phi

$\frac{d}{du} \ln u = \frac{1}{u} du$ if u = cx then $\frac{d}{dx} \ln c x = \frac{1}{cx} \frac{d}{dx} cx = \frac{c}{cx} = \frac{1}{x}$

12. Australopithecus

remember the derivative of ax is a

13. Australopithecus

d/dx ax = ax^0 = a(1) = a

14. Australopithecus

and by the chain rule formula g'(m(x))*m'(x) you should see that a wont leave the function and since it is a log it will cancel out

15. anonymous

ahh.. gotcha. *Click*

16. anonymous

so my mistake was not taking the derivative for both sides, when applying the chain rule. Ln [ c x] u=cx Ln [u] (cx) 1/u c taking derivatives of both terms expand u back again c/cx 1/x simplified.

17. Australopithecus

essentially

18. anonymous

probably not the best way to notate it.. but as a general identity I can say then Ln [c x] = 1/x by chain rule?

19. anonymous

d/dx Ln[c x] = 1/x by chain rule...

20. Australopithecus

yes

21. Australopithecus

Look at the two graphs, note that a constant doesnt change the slope

22. Australopithecus

or rather a constant inside a logarithm doesn't effect the slope o

23. Australopithecus

It just shifts the function

24. perl

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