Question

The derivative of log (sin(logx) (x > 0) ​ibfdd.

Proper answer needed!​

Answer 1

Answer:-

( cos ( ㏒ x ) ) / ( x . sin ( ㏒ x ) )

Explanation:

Let :

y = ㏒ ( sin ( ㏒ x ) )   ,   x > 0

→ ㏒ x = ㏒_e x = ㏑ x

We know :

= ( ㏑ x )' = 1 / x

= y' = 1 / sin ( ㏒ x ) . ( sin ( ㏒ x ) )'

=  y' = 1 / sin ( ㏒ x ) . ( cos ( ㏒ x ) ) . ( ㏒ x )'

= y' = 1 / sin ( ㏒ x ) . ( cos ( ㏒ x ) ) . 1 / x

= y'  = ( cos ( ㏒ x ) ) / ( x . sin ( ㏒ x ) )

Hence we get required answer!

Answer 2

Let , y = Log{Sin(Log(x)}

By chain rule ,

$\begin{gathered}\begin{gathered}\sf \mapsto \frac{dy}{dx} = \frac{1}{Sin \{Log(x) \}} \times \frac{d \{Sin \{Log(x) \} \}}{dx} \\ \\ \sf \mapsto \frac{dy}{dx} =\frac{1}{Sin \{Log(x) \}} \times Cos \{Log(x) \} \times \frac{d \{Log(x) \}}{dx} \\ \\ \sf \mapsto \frac{dy}{dx} = \frac{1}{Sin \{Log(x) \}} \times Cos \{Log(x) \} \times \frac{1}{x} \\ \\ \sf \mapsto \frac{dy}{dx} = \frac{Cos \{Log(x) \}}{x.Sin \{Log(x) \}}\end{gathered}\end{gathered}$

Remmember :

$\begin{gathered}\begin{gathered}\star \sf \: \: \frac{d \{Sin(x) \}}{x} = Cos(x) \\ \\ \star \: \: \sf \frac{d \{Log(x) \}}{dx} = \frac{1}{x}\end{gathered}\end{gathered}$