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In my math class, we are beginning to find derivatives of more complex functions. I’ve been trying questions from my textbook as practice. Here are two of them that I’m trying out:

$y=(\ln x)^2$.
First, we take the power rule. This would make it $2(\ln x)$. Then you multiply it by the derivative of the inside function right? $\ln x$’s derivative is $1/x$. So, multiplied by $1/x$. This gives us $\frac{2\ln x}x$. This doesn’t seem correct so I’m a little confused.

$y=\frac{3}{\sqrt{2x+1}}$

First, we get the denominator to the top. We get $y=3(2x+1)^{-1/2}$. Then, we use the power rule. $y=3(x+1/2)$. Then we multiply by the derivative again. Which is $2$, I believe. This gives us $y=6(2x+1)$. Again, this does not seem correct and I’m confused.

I feel that my mistakes may be from a mix up of steps but I’m not exactly sure where in my process I went wrong.

Ella
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1 Answers1

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First one is correct, for the second one

\begin{eqnarray} \frac{{\rm d}}{{\rm d}x} \frac{3}{\sqrt{2x + 1}} &=& 3\frac{{\rm d}}{{\rm d}x} (2x + 1)^{-1/2} ~~~~\mbox{move constant out} \\ &=& -\frac{3}{2}(2x + 1)^{-3/2}\frac{{\rm d}}{{\rm d}x}(2x + 1) ~~~~\mbox{power rule + chain rule} \\ &=& -\frac{3}{2}(2x + 1)^{-3/2}(2)\\ &=& -\frac{3}{(2x + 1)^{3/2}} \end{eqnarray}

sound wave
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caverac
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