Use The Product Rule To Find The Derivative Of:${ (-7x^8 + 6x 6)(10e X + 5) }$Use { E^{\wedge}x $}$ For { E^x $}$. You Do Not Need To Expand Out Your Answer.
Introduction
In calculus, the product rule is a fundamental concept used to find the derivative of a product of two or more functions. This rule is essential in understanding various mathematical concepts, including optimization problems, physics, and engineering. In this article, we will explore how to use the product rule to find the derivative of a given function.
The Product Rule
The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product, f(x)g(x), is given by:
f(x)g(x) = f'(x)g(x) + f(x)g'(x)
where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.
Applying the Product Rule
Now, let's apply the product rule to find the derivative of the given function:
(-7x^8 + 6x6)(10ex + 5)
To do this, we need to find the derivatives of the two functions separately:
- The derivative of -7x^8 is -56x^7
- The derivative of 6x^6 is 36x^5
- The derivative of 10e^x is 10e^x
- The derivative of 5 is 0
Now, we can apply the product rule:
f(x) = -7x^8 + 6x^6 g(x) = 10e^x + 5
f'(x) = -56x^7 + 36x^5 g'(x) = 10e^x
Using the product rule, we get:
f(x)g(x) = f'(x)g(x) + f(x)g'(x) = (-56x^7 + 36x5)(10ex + 5) + (-7x^8 + 6x6)(10ex)
Simplifying the Expression
Now, let's simplify the expression by distributing the terms:
(-56x^7 + 36x5)(10ex + 5) = -560x7ex - 280x^7 + 360x5ex + 180x^5 (-7x^8 + 6x6)(10ex) = -70x8ex + 60x6ex
Combining the terms, we get:
f(x)g(x) = -560x7ex - 280x^7 + 360x5ex + 180x^5 - 70x8ex + 60x6ex
Conclusion
In this article, we used the product rule to find the derivative of a given function. We applied the product rule by finding the derivatives of the two functions separately and then combining them using the product rule. The resulting expression was simplified by distributing the terms and combining like terms. This example demonstrates the importance of the product rule in calculus and its applications in various mathematical concepts.
Real-World Applications
The product rule has numerous real-world applications in various fields, including:
- Physics: The product rule is used to find the derivative of the kinetic energy of an object, which is essential in understanding the motion of objects.
- Engineering: The product rule is used to find the derivative of the stress on a material, which is essential in designing structures and machines.
- Economics: The product rule is used to find the derivative of the demand for a product, which is essential in understanding the behavior of consumers.
Common Mistakes
When applying the product rule, it's essential to avoid common mistakes, including:
- Forgetting to distribute the terms: Make sure to distribute the terms correctly to avoid errors.
- Not combining like terms: Make sure to combine like terms to simplify the expression.
- Not checking the derivatives: Make sure to check the derivatives of the functions separately to ensure accuracy.
Conclusion
Q&A: Product Rule and Derivatives
Q: What is the product rule in calculus?
A: The product rule is a fundamental concept in calculus that is used to find the derivative of a product of two or more functions. It states that if we have two functions, f(x) and g(x), then the derivative of their product, f(x)g(x), is given by:
f(x)g(x) = f'(x)g(x) + f(x)g'(x)
Q: How do I apply the product rule?
A: To apply the product rule, you need to find the derivatives of the two functions separately and then combine them using the product rule. For example, if we have the function:
f(x) = (2x + 1)(3x - 2)
We need to find the derivatives of the two functions separately:
- The derivative of 2x + 1 is 2
- The derivative of 3x - 2 is 3
Then, we can apply the product rule:
f(x) = (2x + 1)(3x - 2) f'(x) = 2(3x - 2) + (2x + 1)(3) = 6x - 4 + 6x + 3 = 12x - 1
Q: What are some common mistakes to avoid when applying the product rule?
A: Some common mistakes to avoid when applying the product rule include:
- Forgetting to distribute the terms: Make sure to distribute the terms correctly to avoid errors.
- Not combining like terms: Make sure to combine like terms to simplify the expression.
- Not checking the derivatives: Make sure to check the derivatives of the functions separately to ensure accuracy.
Q: How do I simplify the expression after applying the product rule?
A: To simplify the expression after applying the product rule, you need to distribute the terms and combine like terms. For example, if we have the expression:
f(x) = (2x + 1)(3x - 2) + (2x + 1)(4x + 3)
We can simplify the expression by distributing the terms and combining like terms:
f(x) = 6x^2 - 4x + 8x^2 + 11x + 3 = 14x^2 + 7x + 3
Q: What are some real-world applications of the product rule?
A: The product rule has numerous real-world applications in various fields, including:
- Physics: The product rule is used to find the derivative of the kinetic energy of an object, which is essential in understanding the motion of objects.
- Engineering: The product rule is used to find the derivative of the stress on a material, which is essential in designing structures and machines.
- Economics: The product rule is used to find the derivative of the demand for a product, which is essential in understanding the behavior of consumers.
Q: How do I check my work when applying the product rule?
A: To check your work when applying the product rule, you need to:
- Verify that you have found the correct derivatives of the functions separately.
- Verify that you have applied the product rule correctly.
- Simplify the expression by distributing the terms and combining like terms.
- Check that the final expression is correct.
Conclusion
In conclusion, the product rule is a fundamental concept in calculus that is used to find the derivative of a product of two or more functions. By applying the product rule, we can find the derivative of a given function and simplify the expression by distributing the terms and combining like terms. This article provided a comprehensive guide to the product rule, including common mistakes to avoid and real-world applications.