Question

Differentiate: f(theta)=ln(2theta) with respect to theta
(I'm learning derivatives of logarithmic functions)

Answer 1

\[\Large f( \theta )=\ln(2 \theta ) \]
The derivative of ln[f(x)] is \[\Large \frac{ d }{ dx } \ln [f(x)] = f'(x)* \frac{ 1 }{ f(x) }\]

Derivative of 2theta is just two, so:
\[\large \frac{ d }{ dx } \ln(2 \theta ) = 2 * \frac{ 1 }{ 2 \theta }\]

Answer 2

ie, differentiate the inside function (inside the log), and multiply by 1 over the inside function.

Answer 3

so i had a question about derivatives of logarithmic functions in calc.

Answer 4

is it basically inverse? if its ln?

Answer 5

I was watching the lecture but I wasn't sure what exactly it was about..

Answer 6

Yeah kinda.... the simplest way i can think of is like i said above, differentiate the inside function (inside the log), and multiply that by 1 over the inside function.

Answer 7

is that my final answer?

Answer 8

or would it be 1/theta because the two's cancel?

Answer 9

eg \[\large \frac{ d }{ dx} \ln(x^2) = 2x * \frac{ 1 }{ x^2 }\]

Answer 10

and whatever your writing... please continue lol don't mean to interrupt

Answer 11

1/theta because the two's cancel.

Answer 12

ok thanks!

Answer 13

what about this one:

Answer 14

f(t) = ln [sqrt(t^3)-6]

Answer 15

Derivative of the inside function is (change the sqrt to an exponent to simplfify first)
\[\Large \frac{ d }{dt} ((t^3)^{1/2}-6 ) = \Large \frac{ d }{dt} (t^{3/2}-6 ) = \frac{ 3 }{ 2 }t^{1/2}\]

Answer 16

Then multiply that by 1/[sqrt(t^3)-6]

Answer 17

is that 3t^2 / (2t^3 - 12)?

Answer 18

\[\Large f \prime (t) = \frac{ 3 }{ 2 }t^{1/2} * \frac{ 1 }{ \sqrt{t^3}-6 } = \]
if you meant this, then yes \[\Large f \prime (t) = \frac{ 3 t^{1/2} }{ 2\sqrt{t^3}-12 }\]

Answer 19

hmm... its slightly different then what everyone else got.

Answer 20

Well you can write it a little differently, with the exponents and all.

Answer 21

okthanks!

Answer 22

http://www.wolframalpha.com/input/?i=derivative+ln+%5Bsqrt%28t%5E3%29-6%5D+

if you scroll down, you'll see my solution (they just used the sqrt sign instead of a 1/2 exponent).

Answer 23

ok will do.