Question

What is the derivative of ln(lnx^3)

Answer 1

the derivative of ln(lnx^3) is $\frac{1}{x(lnx)}$ .

Step-by-step explanation:

Here we have to find the derivative of ln(lnx^3) , Let's find out:

We have , $ln(lnx^3)$ , Let's differentiate it w.r.t x :

⇒ $\frac{d(ln(lnx^3))}{dx}$

Let $lnx^3 = u$

⇒ $\frac{d(ln(u))}{dx}$

⇒ $\frac{1}{u} (\frac{d(u)}{dx} )$

⇒ $\frac{1}{u} (\frac{d(ln(x^3))}{dx} )$

Let $x^3=v$

⇒ $\frac{1}{u} (\frac{d(ln(v))}{dx} )$

⇒ $\frac{1}{u}(\frac{1}{v} ) (\frac{d(v)}{dx} )$

⇒ $\frac{1}{u}(\frac{1}{v} ) (\frac{d(x^3)}{dx} )$            

⇒ $3x^2(\frac{1}{u})(\frac{1}{v} )$

Putting value of u & v we get:

⇒ $3x^2(\frac{1}{lnx^3})(\frac{1}{x^3} )$

⇒ $\frac{3}{x(lnx^3)}$    { $lnx^n = n(lnx)$  }

⇒ $\frac{3}{3x(lnx)}$

⇒ $\frac{1}{x(lnx)}$

Therefore , the derivative of ln(lnx^3) is $\frac{1}{x(lnx)}$ .