MHB Derivative of 2^x - Is it 2^x ln(x) or ln(2)?

[tmt1]

Hi,

In my notes I put down that

y=2^x
y'=2^x ln(x)

However, I seem to remember that it is in fact

y'=2^x ln(2)

Which one is correct?

Thanks,

Tim

 

[Ackbach]

\begin{align*}
y&=2^x \\
\ln(y)&= \ln(2^x) \\
\ln(y)&= x \ln(2) \\
\frac{y'}{y}&=\ln(2) \\
y'&= y \ln(2) \\
y'&= \ln(2) \, 2^x.
\end{align*}

 

[MarkFL]

Another way to view it (although I favor Ackbach's method) is:

$$y=2^x=e^{\ln\left(2^x \right)}=e^{x\ln(2)}$$

And so:

$$y'=e^{x\ln(2)}\cdot\ln(2)=\ln(2)2^x$$