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Problem:

Differentiate with respect to $x$: $\sin^2(\ln(x^2))$

My book's solution:

$$\frac{d}{dx}(\sin^2(\ln(x^2)))$$

$$2\sin\ln x^2.\cos\ln x^2.\frac{1}{x^2}.2x$$

$$\frac{2}{x}\sin(2\ln x^2)$$

$$\frac{2}{x}\sin(4\ln x)\tag{1}$$

Question:

  1. Isn't $(1)$ wrong: $x$ could be a negative number, and then we wouldn't be able to apply logarithm properties?
tryingtobeastoic
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    Yes you are right. – Z Ahmed Oct 12 '21 at 15:30
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    It should be $\frac 2x \sin(4 \ln |x|)$ instead of what's written above. – Sarvesh Ravichandran Iyer Oct 12 '21 at 15:32
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    Notice everything is fine until the last step, and when taking the log of the square, we really can just pop an absolute value in there, giving @TeresaLisbon's answer. – Cade Reinberger Oct 12 '21 at 15:35
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    $+1$ though can you please try to know what happens when you'll take $-$ve values and giving you the complex numbers. (Just for the sake of knowledge) and $\sin()$ of complex and all, Well @TeresaLisbon is right✌️ –  Oct 12 '21 at 15:50

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