Help with log differentiation

Question

The question is if $y = n^t$ then find $\frac{d(\ln(y))}{dt}$

I keep getting $\frac{n}{t}$ as the answer while according to the book its $\ln(n)$

Answer 1

If $y = n^t$, then $\ln(y) = \ln(n^t) = t\cdot\ln(n)$. Then when we differentiate with respect to $t$, we treat $n$ as a constant (so $\ln(n)$ is a constant too!), and we get $\ln(n)$.

Answer 2

If $y=n^t$, then $\ln y=t\ln n$ and therefore$$\frac{d\ln y}{dt}=\ln n.$$

Answer 3

$y = n^t = e^{t \ln(n)}$

So $\ln(y) = t \, \ln(n)$

So ${{d \, \ln(y)} \over dt} = {d \over dt} \ln(n) \, t = \ln(n)$