Answer:
To differentiate the given function, we can use the product rule, which states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.
Let's differentiate the function step by step:
Given function: y = (6x^4 - x + 1)(-x^5 + 1)
Using the product rule, we have:
y' = (6x^4 - x + 1)(-x^5 + 1)' + (6x^4 - x + 1)'(-x^5 + 1)
Now, let's find the derivatives of each term:
The derivative of (-x^5 + 1) is (-5x^4 + 0) = -5x^4.
The derivative of (6x^4 - x + 1) is (24x^3 - 1).
Substituting these derivatives back into the equation, we have:
y' = (6x^4 - x + 1)(-5x^4) + (24x^3 - 1)(-x^5 + 1)
Simplifying further:
y' = -30x^8 + 5x^5 - 6x^4 + x - 24x^3 + 1
Therefore, the derivative of the given function y = (6x^4 - x + 1)(-x^5 + 1) is y' = -30x^8 + 5x^5 - 6x^4 + x - 24x^3 + 1.
Explanation: