Question

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

Answer 1

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

Answer 2

Assuming it is log based 10

\[\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{\ln(a)*u(x)}\]

Answer 3

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 ???

Answer 4

its ln(x)?

Answer 5

oh assuming oh assuming base E

Answer 6

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

Answer 7

see
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/der_common_functions/v/proof-d-dx-ln-x-1-x

Answer 8

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)

Answer 9

the constant cancels out

Answer 10

For the derivative

Answer 11

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

Answer 12

remember the derivative of

ax is a

Answer 13

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

Answer 14

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

Answer 15

ahh.. gotcha. *Click*

Answer 16

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.

Answer 17

essentially

Answer 18

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?

Answer 19

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

Answer 20

yes

Answer 21

Look at the two graphs,

note that a constant doesnt change the slope

Answer 22

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

Answer 23

It just shifts the function

Answer 24

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