Derivatives

[Malga1968]

If I can please have some help.

Finding the derivative of In(3x):

Would it be 1/3x x 3 = 1/x. So my answer would be 1/x

Would that apply with any function as In(5x). My answer would also be 1/x.

So would it be for any positive number for x it would always = 1/x

Am I doing this correctly.

Thank you for the help.
Malga

 

[galactus]

Yes, in general, \(\displaystyle \frac{d}{dx}[ln(|a|x)]=\frac{1}{x}\). 'a' can be negative also.

\(\displaystyle a\neq{0}\)

 

[Malga1968]

Thank you galactus. Malga

 

[soroban]

Hello, Malga1968!

\(\displaystyle \text{Find the derivative of: }\;\ln(3x)\)

\(\displaystyle \text{Would it be: }\;\frac{1}{3x}\cdot3 \:= \:\frac{1}{x}\)

\(\displaystyle \text{Would that apply with any function as: }\;\ln(5x)\)
. . \(\displaystyle \text{My answer would also be: }\;\frac{1}{x}\)

Yes, you are right!

When I first ran into that, I was puzzled . . . It didn't seem right.
Then I saw WHY it was true.

\(\displaystyle \text{Consider: }\;y \;=\;\ln(7x)\)

\(\displaystyle \text{Recall that: }\;y \:=\:\ln(7x) \;=\;\ln(7) + \ln(x)\)

Now differentiate it . . .