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We can show this using the fact that \sin x \sim x for small values of x; precisely, we have the inequality \frac 1 2 x \le \sin x for all x \in [0,1]; this leads to \ln\left (\frac {x} {2}\right) \le \ln \sin x ... Consider c = e^ {2 \pi}. Then, f (nc) = f (n) for all n.

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How do you find \displaystyle\frac { {\left. {d} {y}\right.}} { {\left. {d} {x}\right.}} by implicit differentiation given \displaystyle {\ln { {y}}}= {\sin { {x}}} ?

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Derivative of y = ln(sin x)

I=\underbrace {\int_0^ {\frac {\pi} {2}}\ln\sin x\mathrm {d}x}_ {x\rightarrow \pi-x}=\int_ {\pi}^ {\frac {\pi} {2}}\ln\sin (\pi-x)\mathrm {d} (\pi-x)=\int_ {\frac {\pi} {2}}^ {\pi}\ln\sin x\mathrm {d}x\\

In this video, let us learn how to find the derivative of ln (sin x) [natural logarithm of sin x]. If y=ln (sinx), then find dy/dx. ...more

The f (x)=ln (sin (x)) is Parity: Neither even nor odd<br/>Inverse: arcsin (e^x)