Derivative of 3^(x/2)
By James Williams
Im having trouble with this derivative: $3^{x/2}$
I know that the derivative of $c^x$ is $c^x\ln c$.
Is $3^{x/2} = 3^{x/2}\ln 3$ correct?
$\endgroup$2 Answers
$\begingroup$The equals sign should only be used for equality. For example, when finding the derivative of $x^2$, please do not write "$x^2 = 2x$". It does not make sense. If $f(x)=x^2$, then $f'(x)=2x$ would be a better way to state the derivative.
No, $3^{x/2}\ln 3$ is not correct, but your formula for the derivative of $c^x$ is correct, and can be obtained using the chain rule by observing that $c^x=e^{x\ln c}$. And if you know the derivative of $f(x)=3^x$, then you can find the derivative of $g(x)=f(x/2)$ using the chain rule. Alternatively, note that $3^{x/2}=(\sqrt{3})^x$.
$\endgroup$ 6 $\begingroup$Well, almost. It's a composite function, so you need to use the chain rule; multiply with the derivative of $x/2$. So the correct solution is $$ (3^{x/2})^' = \frac{1}{2} \cdot \text{ln}3 \cdot 3^{x/2}.$$
$\endgroup$
