Math Problem Statement
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
Question content area bottom
Part 1
y prime equals nothing
Solution
To differentiate the function , we'll use the product rule. The product rule states that if , then:
Here, we can identify:
Step 1: Differentiate and
First, let's find the derivatives of and :
Step 2: Apply the Product Rule
Now, apply the product rule:
Substitute the derivatives and the original functions:
Step 3: Expand the Expression
Now, we'll expand the expression to simplify it:
Expand the first term:
Expand the second term:
Step 4: Combine Like Terms
Finally, add the expanded expressions together:
Combine like terms:
So, the derivative is:
Would you like further details or have any questions about these steps?
Here are some related questions to expand your understanding:
- How does the product rule change if one of the functions is a constant?
- What is the significance of the higher-order terms in the derivative of a product?
- How would the derivative change if the original function was divided instead of multiplied?
- How does the derivative behave as approaches infinity?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Differentiation
Product Rule
Polynomial Functions
Formulas
Product Rule for differentiation
Theorems
-
Suitable Grade Level
Advanced High School
Related Recommendation
Derivative of f(x) = (-x^5 + 6x^2 - 3)(5x^4 - 4x) using the Product Rule
Find the Derivative of f(x) = x^4 (x^9 - 8) Using Product Rule and Simplification
Differentiate Function Using Product Rule: (4x - 2)^2(3 - x^5)^2
Derivative of f(x) = (x^2 - 4x + 2)(5x^3 - x^2 + 4) Using Product Rule
Derivative of (4x^2 - 5)(4x^4 - 7x) Using the Product Rule