Differentiate following.

Solution :It requires the product rule & each derivative in the product rule will need a chain rule application as well.

T ′ ( x ) =1/1+(2x)² (2) (1-3x²)^(1/3) +tan^-1(2x)(1/3)(1-3x²)^(-2/3)(-6x)

= 2(1 - 3x² )^(1/3)  /(1+(2x)2 - 2(1 - 3x² )^-(2/3)  tan ^-1 ( 2x )

We know that,

d (tan ^-1 x ) / dx =  (1/(1+x²)

While doing the chain rule with this we remember that we've got to leave the inside function

alone. That means that where we have the x²  in the derivative of tan ^-1 x we will have to have (inside function )² .