To differentiate the function $y = (6x^4 - x + 4)(-x^5 + 9)$, we'll use the product rule. The product rule states that if $y = u(x) \cdot v(x)$, then:

$y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)$

Here, we can identify:

$u(x) = 6x^4 - x + 4$ $v(x) = -x^5 + 9$

Step 1: Differentiate $u(x)$ and $v(x)$

First, let's find the derivatives of $u(x)$ and $v(x)$:

$u'(x) = \frac{d}{dx}(6x^4 - x + 4) = 24x^3 - 1$

$v'(x) = \frac{d}{dx}(-x^5 + 9) = -5x^4$

Step 2: Apply the Product Rule

Now, apply the product rule:

$y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)$

Substitute the derivatives and the original functions:

$y' = (24x^3 - 1)(-x^5 + 9) + (6x^4 - x + 4)(-5x^4)$

Step 3: Expand the Expression

Now, we'll expand the expression to simplify it:

Expand the first term:

$(24x^3 - 1)(-x^5 + 9) = 24x^3(-x^5) + 24x^3(9) - 1(-x^5) - 1(9)$ $= -24x^8 + 216x^3 + x^5 - 9$

Expand the second term:

$(6x^4 - x + 4)(-5x^4) = 6x^4(-5x^4) - x(-5x^4) + 4(-5x^4)$ $= -30x^8 + 5x^5 - 20x^4$

Step 4: Combine Like Terms

Finally, add the expanded expressions together:

$y' = (-24x^8 + 216x^3 + x^5 - 9) + (-30x^8 + 5x^5 - 20x^4)$

Combine like terms:

$y' = (-24x^8 - 30x^8) + (x^5 + 5x^5) - 20x^4 + 216x^3 - 9$ $y' = -54x^8 + 6x^5 - 20x^4 + 216x^3 - 9$

So, the derivative is:

$y' = -54x^8 + 6x^5 - 20x^4 + 216x^3 - 9$

Would you like further details or have any questions about these steps?

Here are some related questions to expand your understanding:

1. How does the product rule change if one of the functions is a constant?
2. What is the significance of the higher-order terms in the derivative of a product?
3. How would the derivative change if the original function was divided instead of multiplied?
4. How does the derivative behave as $x$ approaches infinity?
5. How would you apply the quotient rule to a similar problem?

Tip: When expanding polynomials during differentiation, be careful with the signs, especially when negative terms are involved.