explain the steps of using the chain rule to find `dy/dx` of `log(sin^6x)`

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ishpiro eNotes educator| Certified Educator

Chain rule states that the derivative of the composition function `f(g(x))` is `(df)/(dg) * (dg)/(dx)` . This also applies to the composition of three functions `f(g(h(x)))` . The derivative `(df)/dx = (df)/(dg)*(dg)/(dh)*(dh)/(dx)` .

The function`log(sin^6(x))` is a composition of three function: log of the 6th power of the sine of x. So its derivative with respect to x will be the product of three derivatives:

derivative of log is `1/(ln10*(sin^6(x)))`

derivative of `sin^6(x)` is`6sin^5(x)`

derivative of sin(x) is cos(x).

Thus the derivative of `log (sin^6(x))` is

`1/(ln10* (sin^6*(x))) * 6sin^5(x)*cos(x)`

This can be simplified to `(6cos(x))/(ln10*sin(x))` . 

The derivative of `log(sin^6(x))` is `` `(6cos(x))/(ln10*sin(x))` .