Differentiate the function.

y equals left parenthesis 6 x Superscript 4 Baseline minus x plus 4 right parenthesis left parenthesis negative x Superscript 5 Baseline plus 9 right parenthesis

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Part 1

y prime equals nothing
  
   

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.

Learn how to differentiate the function (6x^4 - x + 4)(-x^5 + 9) using the product rule. This problem involves polynomial functions and demonstrates the application of the product rule in calculus. Suitable for advanced high school students and anyone studying calculus.

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