Differentiate the Function

Question

(5x-4)^5(4-x^3)^5

Differentiate the function.

Answer 1

Let y = (5x - 4)^5(4 - x^3)^5.

Recall the product rule of derivatives,

d/dx(uv)=uv'+vu'.

u = (5x - 4)^5

u' = 25(5x - 4)^4

v = (4 - x^3)^5

v' = 5(4 - x^3)^4( - 3x^2 )

dy/dx = (5x - 4)^5 * [ 5(4 - x^3)^4( - 3x^2 ) ] + (4 - x^3)^5 * [ 25(5x - 4)^4 ]

             = - 15x^2(5x - 4)^5(4 - x^3)^4 + 25(4 - x^3)^5(5x - 4)^4

             = 5(5x - 4)^4(4 - x^3)^4[ 5(4 - x^3) - 3x^2(5x - 4) ]

             = 5(5x - 4)^4(4 - x^3)^4[ 20 - 5x^3 - 15x^3 + 12x^2 ]

             = 5(5x - 4)^4(4 - x^3)^4[ 20 - 20x^3 + 12x^2 ]

             = - 20(4 - 5x)^4(x^3 - 4)^4(5x^3 - 3x^2 - 5).

dy/dx = - 20(4 - 5x)^4(x^3 - 4)^4(5x^3 - 3x^2 - 5).