Question

3. If y = log10sin x,
Dy/DX
(a) cotx
b) cot x .log10e
(c) tan x
(d) log10 cos x​

Answer 1

Answer:

option (b) $\cot x.\log _{10}e$ is the differential to the function $y=\log _{10}\sin x$.Step-by-step explanation:

The question is to differentiate the function $y=\log _{10}\sin x$

To solve this, firstly we want to change the base of log, that is change base 10 in to base ' e '. For that we are using the property of log. The property of log is given as follows,

  $\log _{a}b=\dfrac{\log _{c}b}{\log _{c}a}$

Therefore,

            $y=\dfrac{\log _{e}\sin x}{\log _{e}10}$

Now we can differentiate directly,

      $\begin{array}{l}\dfrac{dy}{dx}=\dfrac{1}{\log_e10}\cdot\dfrac{d}{dx}\left(\log_e\sin x\right)\\\\\ \ \ \ \ \ \ =\dfrac{1}{\log_e10}\cdot\dfrac{1}{\sin x}\cdot\cos x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ <\dfrac{\cos x}{\sin x}=\cot x>\\\\\ \ \ \ \ \ \ \ =\dfrac{\cot x}{ log _{c}10}\end{array}$

If we take the denomenator to numerator, it will become as follows,

       $\dfrac{dy}{dx}=\cot x.\log _{10}e$

Hence option (b) $\cot x.\log _{10}e$ is the differential to the function $y=\log _{10}\sin x$.