Question

Question #f1456

Answer 1

`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)`