Find the derivative of the function. f(x) = (2x - 2)⁴ (x²+x+1)⁵.

Solution:

Given function f(x) = (2x − 2)⁴ (x² + x + 1)⁵

To find the derivative of f(x), we use the product rule of differentiation for f(x)= g(x).h(x), we have

f′(x) = g′(x).h(x) +g(x).h′(x)

Here, g(x)=(2x-2)⁴ ; h(x) = (x²+ x +1)⁵

Hence, g′(x) = 4(2x-2)³(2) = 8(2x-3)³

h′(x) = 5( x²+ x +1)⁴(2x+1)

f′(x)= {8(2x-2)³}{(x²+ x +1)⁵} + {(2x-3)⁴}{ 5( x²+ x +1)⁴(2x+1)}

=(2x-2)³(x²+ x +1)⁴{8(x²+ x +1) + 5(2x-3)(2x+1)}

=(2x-2)³(x²+ x +1)⁴ {8x²+8x+8+20x² -10x -10}

=(2x-2)³(x²+ x +1)⁴ {28x²-2x-2}

Find the derivative of the function. f(x) = (2x - 2)⁴ (x²+x+1)⁵.

Summary:

The derivative of the function. f(x) = (2x - 2)⁴ (x²+x+1)⁵ =(2x-2)³(x²+ x +1)⁴ {28x²-2x-2}