Question #f1456

1 Answer
Jan 28, 2018

#cot(x)/ln(10)#

Explanation:

Remember the change of base formula: #log_ab=(ln(b))/(ln(a))#.

Here, #a=10,# and #b=sin(x)#.

So input: #d/dx((ln(sin(x)))/(ln(10)))#

Take the constant out. Basically, the part that does not affect the variable #x#. For example, while solving for #d/dx2x^2#, you could take the #2# out as it would not affect the final value.

#1/(ln(10))d/dxln(sin(x))#.

Now apply the chain rule. #(df(u))/dx=(df)/(du)*(du)/dx#.

Here, #f=lnu#, and #u=sin(x)#.

#d/(du)lnu=1/u#

#d/dxsin(x)=cos(x)#

So #(df(u))/dx=1/u*cos(x)=cos(x)/u#.

But remember, #u=sin(x)#

So it becomes #cos(x)/sin(x)=cot(x)#

Multiply this by the constant, #1/ln(10)*cot(x)=cot(x)/ln(10)#

The derivative of #log(sin(x))# is #cot(x)/ln(10)#