Question

What is the derivative of `x^3 * (2/3x^2 -1)^4`?

Answer 1

`176/81 x^10 -32/3 x^8 +56/3 x^6 -40/3 x^4 +3x^2`

Explanation:

Expand the expression as

`x^3 [ (2/3 x^2)^4 +4 (2/3 x^2)^3 (-1) +6 (2/3 x^2)^2 (-1)^2 +4(2/3 x^2) (-1)^3 +(-1)^4]`

=`16/81 x^11 -32/27 x^9 +24/9 x^7 -8/3 x^5 +x^3`

Its first derivative would be

`176/81 x^10 -32/3 x^8 +56/3 x^6 -40/3 x^4 +3x^2`

Alternatively, product rule of differentiation can also be used.