Using the Log Law `log\ a^n= n\ log\ a`, we can write:

*y* = ln(2*x*^{3} − *x*)^{2} = 2 ln(2*x*^{3} − *x*)

Put

`u = 2x^3 − x`

so

`u’ = 6x − 1`

This gives us:

`(dy)/(dx)=2(6x^2-1)/(2x^3-x`

`x ≠ ±sqrt(0.5)`,

`x ≠ 0`

**NOTE:** We need to be careful with the domain of this solution, as it is only correct for certain values
of *x*.

The graph of *y* = ln(2*x*^{3} - *x*)^{2} is defined for all *x* except

` Âħsqrt(0.5), 0`

Its graph is as follows:

The graph of *y* = 2 ln(2*x*^{3} -
*x*), however, is only defined for a more limited
domain (since we cannot have the logarithm of a negative
number.)

So we can only have *x* in the range `-sqrt 0.5 < x < 0` and `x > sqrt0.5.`

So when we find the differentiation of a logarithm using the shortcut given above, we need to be careful that the domain of the function and the domain of the derivative are stated.