Question

Find the derivative?

`y=(x^2 - 5x)^3(8x-7)^2`

Answer 1

`512x^7 -7504x^6 +39174x^5 -85675x^4 +70700x^3 -18375x^2`

Explanation:

This derivative can be found by using product rule or more simply by multiplying the factors and then differentiating term by term. Thus the given expression is equivalent to

`(x^6 -15x^5 +75x^4 -125x^3)(64x^2 -112x +49)`

Or, `64x^8 -960x^7 +4800x^6-8000x^5 -112x^7 +1680x^6 -8400x^5 +14000x^4 +49x^6 - 735x^5 +3675x^4-6125x^3`

Or, `64x^8 -1072x^7 +6529x^6 -17135x^5 +17675x^4 -6125x^3`

Now differentiate term by term

`512x^7 -7504x^6 +39174x^5 -85675x^4 +70700x^3 -18375x^2`

Answer 2

Explained below

Explanation:

`3(x−5)2⋅(x3)(8x−7)2]+[3(x−5)3⋅x2(8x−7)2]+[16(x−5)3⋅x3⋅(8x−7)]`
This expression when simplified would also lead to the same answer, as shown below:
(A bit of laborious work is required)

=`3(x^2 -10x +25)x^3 (64x^2-112x+49) +3(x^3 - 15x^2 +75x -125)x^2 (64x^2 -112x +49)+16(x^3-15x^2 +75x -125) (8x^4-7x^3)`

=`(3x^5 -30x^4 +75x^3)(64x^2 -112x +49)+(3x^5 -45x^4 +225x^3 -375x^2)(64x^2 -112x +49) +(16x^3 -240x^2+1200x -2000)(8x^4 -7x^3)`

=#192x^7 -1920x^6 +4800x^5 -336x^6 + 3360x^5 -8400x^4 + 147x^5 -1470x^4 +3675x^3#

#+192x^7 -2880x^6 +14400x^5 -24000x^4 -336x^6 +5040x^5 -25200x^4 +42000x^3 +147x^5 - 2205x^4 +11025x^3 -18375x^2 +128x^7 -1920x^6 +9600x^5 -16000x^4 -112x^6 +1680x^5 -8400x^4 +14000x^3#

=`512x^7 -7504x^6 +39174x^4 -85675x^4 +70700x^3 -18375x^2`