Question

What is the derivative of `f(x)= (2x-3)^3 (x+1)^2`?

Answer 1

`f'(x)=40x^4-80x^3-30x^2+90x`

Explanation:

Note: this is simply an alternative method to Steve M's version which uses the Product and Chain Rules (and was what was probably intended)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given: `f(x)=(2x-3)^3(x+1)^2`

`(2x-3)^3` can be easily expanded using `color(brown)("Pascal's Triangle")` as:
`(2x-3)^3 = color(brown)1 * (2x)^3 + color(brown)3 * (2x)^2 * (-3)^1 +color(brown)3 * (2x)^1 * (-3)^2 + color(brown)1 * (-3)^3`
`color(white)((2x-3)^3)=8x^3-36x^2+54x-27`

and the expansion of `(x+1)` is trivially common:
`(x+1)^2=x^2+2x+1`

`f(x)=(2x-3)^3(x+1)^2` can therefore be expanded with minimal effort (I will use tabular multiplication for demonstration purposes)
and then the derivative can be found by by noting that the derivative of a sum of terms is simply the sum of the derivatives of the individual terms (and applying the exponent rule for derivatives)

#{: (,ul(xx)," | ",ul(8x^3),ul(-36x^2),ul(+54x),ul(-27)), (,x^2," | ",8x^5,-36x^4,+54x^3,-27x^2), (,+2x," | ",+16x^4,-72x^3,+108x^2,-54x), (,ul(+1),ul(" | "),ul(+8x^3),ul(-36x^2),ul(+54x),ul(-27)), (f(x)=,8x^5,-20x^4,-10x^3,+45x^2,,-27), (color(white)("xx"),,,,,,), (f'(x)=,40x^4,-80x^3,-30x^2,+90x,,) :}#

This is simply an expansion of Steve M's version (and you should look at it, if only as an application of the Product and Chain Rules)

Answer 2

`f'(x) = 10x(2x-3)^2(x+1)`

Explanation:

We seek the derivative of:

`f(x) = (2x-3)^3(x+1)^2`

We can apply the product rule so that:

`f'(x) = (2x-3)^3 \ (d/dx(x+1)^2) + (d/dx(2x-3)^3) \ (x+1)^2`

And by application of the chain rule, we have:

`d/dx(x+1)^2 = 2(x+1) \ (d/dx(x+1))`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 2(x+1)`

Similarly:

`d/dx(2x-3)^3 = 3(2x-3)^2 \ d/dx(2x-3)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 3(2x-3)^2 (2)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 6(2x-3)^2`

Thus we have:

`f'(x) = (2x-3)^3 \ 2(x+1) + 6(2x-3)^2 \ (x+1)^2`

`\ \ \ \ \ \ \ \ \= (2x-3)^2(x+1){ 2(2x-3) + 6 (x+1) }`

`\ \ \ \ \ \ \ \ \= (2x-3)^2(x+1)(4x-6 + 6x+6)`

`\ \ \ \ \ \ \ \ \= (2x-3)^2(x+1)(10x)`

`\ \ \ \ \ \ \ \ \= 10x(2x-3)^2(x+1)`