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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?

2 Answers 2

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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 $$ (3^{x/2})^' = \frac{1}{2} \cdot \text{ln}3 \cdot 3^{x/2}.$$

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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 $3^{x/2}=(\sqrt{3})^x$.

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    Jonas, I think he meant his statement to be read as (derivative of $c^x) = c^x \log c$, which would fall in line with what you are thinking.2011-01-19
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    @user1736: I was writing the first part in response to the semantics of "Is $3^{x/2} = 3^{x/2}\ln 3$ correct?" I might not have made the appeal if it had said "Is the derivative of $3^{x/2} = 3^{x/2}\ln 3$ correct?", even though that may not be ideal.2011-01-19
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    @user1736: I agree that was what OP was thinking, but the point of using the equals sign incorrectly is important. You can easily get fooled looking back, as in the last line which could be seen as requiring Ln3=12011-01-19
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    Who is OP that keeps getting mentioned?2011-01-19
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    @Arjang In this context OP stands for "original poster".2011-01-19
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    @Jonas, ah, I see what you mean now. I think my brain just processed the "derivative of " part again on its own in the last statement. Sorry about that.2011-01-19