The derivative of natural log?

y=-ln (3/2x)

2 Answers
Jun 18, 2018

y' = - frac(1)(x)

Explanation:

We have: y = - ln(frac(3)(2) x)

We must use the chain rule to differentiate y.

Let u = frac(3)(2) x Rightarrow u' = frac(3)(2) and v = - ln(u) Rightarrow v' = - frac(1)(u):

Rightarrow y' = u' cdot v'

Rightarrow y' = frac(3)(2) cdot - frac(1)(frac(3)(2) x)

Rightarrow y' = frac(3)(2) cdot - frac(2)(3 x)

therefore y' = - frac(1)(x)

Jun 18, 2018

-1/x

Explanation:

Remember the form:

(d)/(dx)lnf(x) = f^'(x)/f(x)

y = -ln((3x)/2)

y = ln(2/(3x))

\therefore(dy)/(dx) = -2/(3x^2) \div 2/(3x)

(dy)/(dx) = -1/x

Hope that makes sense!