Q.Differentiate $y=e^{3x^2}$ with respect to $x$.

Convert to natural logs, differentiate with the quotient rule, and evaluate at $x=e$ where $\ln x=1,\ \ln(\ln x)=0$.

Differentiating a change-of-base logarithm is a CBSE/NCERT Class 12 continuity and differentiability exercise.

Using change of base, $f(x)=\log_x(\log_e x) = \dfrac{\ln(\ln x)}{\ln x}$.

Let $N=\ln(\ln x)$ and $D=\ln x$. Then

$N' = \frac{1}{\ln x}\cdot\frac1x = \frac{1}{x\ln x}, \qquad D' = \frac1x.$

Quotient rule:

$f'(x) = \frac{N'D - ND'}{D^2} = \frac{\dfrac{1}{x\ln x}\cdot\ln x - \ln(\ln x)\cdot\dfrac1x}{(\ln x)^2} = \frac{\dfrac1x\big(1-\ln(\ln x)\big)}{(\ln x)^2}.$

At $x=e$: $\ln x = 1$ and $\ln(\ln x) = \ln 1 = 0$, so

$f'(e) = \frac{\tfrac1e(1-0)}{1^2} = \frac{1}{e}.$