Derivative of log(sin x) by First Principle

If f(x) is a function of the real variable x, then its derivative by the first principle of the derivative is given by

$f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$ $\cdots (i)$

Here ${}^{\prime }$ denotes the derivative. In this post, we will find the derivative of \log(\sin x) by the first principle of derivatives.

Derivative of log(sinx) from first principle

Let $f(x)=\log (\sin x)$.  By the above rule (i) of the first principle of derivatives, we obtain the derivative of $\log (\sin x)$ as follows:

${\displaystyle \frac{d}{dx}}(\log (\sin x))=(\log \sin x{)}^{\prime }$$=\underset{h\to 0}{lim}{\displaystyle \frac{\log (\sin (x+h))-\log (\sin x)}{h}}$

Step 1: We will apply the formula of $\log m-\log n=\log (m/n)$. By doing so we get that

$=\underset{h\to 0}{lim}{\displaystyle \frac{\log (\frac{\sin (x+h)}{\sin x})}{h}}$

Step 2: Applying the trigonometric formula of sin(a+b)=sin a cos b+cos a sin b, the above is

$=\underset{h\to 0}{lim}$ ${\displaystyle \frac{\log ({\displaystyle \frac{\sin x\cos h+\cos x\sin h}{\sin x}})}{h}}$

$=\underset{h\to 0}{lim}$ ${\displaystyle \frac{\log ({\displaystyle \frac{\sin x\cos h}{\sin x}}+{\displaystyle \frac{\cos x\sin h}{\sin x}})}{h}}$

$=\underset{h\to 0}{lim}$ ${\displaystyle \frac{\log (\cos h+\cot x\sin h)}{h}}$ as we know that $\frac{\cos x}{\sin x}=\cot x$.

Step 3: As $\cos h$ tends to $1$ when $h\to 0$, from the step 2 we obtain the derivative of $\log \sin x$ is

$=\underset{h\to 0}{lim}$ $[{\displaystyle \frac{\log (1+\cot x\sin h)}{\cot x\sin h}}$ $\times {\displaystyle \frac{\cot x\sin h}{h}}]$

Let $t=\cot x\sin h$. Then $t\to 0$ when $h\to 0$.

$=\underset{t\to 0}{lim}$ ${\displaystyle \frac{\log (1+t)}{t}}$ $\times \underset{h\to 0}{lim}{\displaystyle \frac{\cot x\sin h}{h}}$

$=1\times \cot x\underset{h\to 0}{lim}{\displaystyle \frac{\sin h}{h}}$ as the limit of log(1+t)/t is 1 when t tends to zero.

$=\cot x\times 1$ as $\underset{h\to 0}{lim}{\displaystyle \frac{\sin h}{h}}=1$

$=\cot x$

Thus the derivative of log(sin x) is cot x, and this is obtained by the first principle of derivatives.

Also Read:

Derivative of root sin x from first principle

Derivative of root cos x from first principle

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