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Mar 12, 2025 by ADMIN 187 views

**Using the Product Rule to Find the Derivative of a Complex Function**

Introduction

In calculus, the product rule is a fundamental concept used to find the derivative of a function that is the product of two or more functions. This rule is essential in understanding how to differentiate complex functions, which are common in various fields such as physics, engineering, and economics. In this article, we will use the product rule to find the derivative of a complex function, specifically $\left(-4 x^8 + 4 x^9\right)\left(4 e^x - 8\right)$ .

Understanding the Product Rule

The product rule states that if we have a function of the form $f(x) = u(x)v(x)$ , where $u(x)$ and $v(x)$ are both functions of $x$ , then the derivative of $f(x)$ with respect to $x$ is given by:

f'(x) = u'(x)v(x) + u(x)v'(x)

This rule can be extended to functions with more than two factors. In our case, we have a function with two factors, so we will use the product rule to find its derivative.

Applying the Product Rule

To find the derivative of $\left(-4 x^8 + 4 x^9\right)\left(4 e^x - 8\right)$ , we will first identify the two factors:

u(x) = -4 x^8 + 4 x^9

v(x) = 4 e^x - 8

Next, we will find the derivatives of $u(x)$ and $v(x)$ with respect to $x$ :

u'(x) = -32 x^7 + 36 x^8

v'(x) = 4 e^x

Now, we can apply the product rule to find the derivative of the original function:

f'(x) = u'(x)v(x) + u(x)v'(x)

f'(x) = (-32 x^7 + 36 x^8)(4 e^x - 8) + (-4 x^8 + 4 x^9)(4 e^x)

Simplifying the Derivative

To simplify the derivative, we can expand the product and combine like terms:

f'(x) = (-32 x^7 + 36 x^8)(4 e^x - 8) + (-4 x^8 + 4 x^9)(4 e^x)

f'(x) = -128 x^7 e^x + 144 x^8 e^x - 256 x^7 + 288 x^8 - 16 x^8 e^x + 16 x^9 e^x

f'(x) = -128 x^7 e^x + 128 x^8 e^x - 256 x^7 + 272 x^8 + 16 x^9 e^x

Conclusion

In this article, we used the product rule to find the derivative of a complex function, specifically $\left(-4 x^8 + 4 x^9\right)\left(4 e^x - 8\right)$ . We identified the two factors, found their derivatives, and applied the product rule to find the derivative of the original function. The resulting derivative was simplified to make it easier to understand and work with. The product rule is a powerful tool in calculus that allows us to find the derivative of complex functions, and it is essential in understanding various concepts in mathematics and other fields.

Example Use Cases

The product rule has many 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 understanding the behavior of materials under different loads.
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.

Tips and Tricks

When using the product rule, it is essential to:

Identify the factors: Clearly identify the two factors of the function and their derivatives.
Apply the product rule: Use the product rule to find the derivative of the original function.
Simplify the derivative: Simplify the derivative to make it easier to understand and work with.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the product rule, a fundamental concept in calculus.

Q: What is the product rule?

A: The product rule is a formula used to find the derivative of a function that is the product of two or more functions. It states that if we have a function of the form $f(x) = u(x)v(x)$ , where $u(x)$ and $v(x)$ are both functions of $x$ , then the derivative of $f(x)$ with respect to $x$ is given by:

f'(x) = u'(x)v(x) + u(x)v'(x)

Q: When do I use the product rule?

A: You use the product rule when you need to find the derivative of a function that is the product of two or more functions. This can be a complex function, such as $\left(-4 x^8 + 4 x^9\right)\left(4 e^x - 8\right)$ .

Q: How do I apply the product rule?

A: To apply the product rule, you need to:

Identify the factors: Clearly identify the two factors of the function and their derivatives.
Apply the product rule: Use the product rule to find the derivative of the original function.
Simplify the derivative: Simplify the derivative to make it easier to understand and work with.

Q: What are some common mistakes to avoid when using the product rule?

A: Some common mistakes to avoid when using the product rule include:

Not identifying the factors: Failing to clearly identify the two factors of the function and their derivatives.
Not applying the product rule correctly: Failing to use the product rule to find the derivative of the original function.
Not simplifying the derivative: Failing to simplify the derivative to make it easier to understand and work with.

Q: Can I use the product rule with more than two factors?

A: Yes, you can use the product rule with more than two factors. The product rule can be extended to functions with more than two factors. For example, if we have a function of the form $f(x) = u(x)v(x)w(x)$ , where $u(x)$ , $v(x)$ , and $w(x)$ are all functions of $x$ , then the derivative of $f(x)$ with respect to $x$ is given by:

f'(x) = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)

Q: How do I know when to use the product rule versus the chain rule?

A: The product rule and the chain rule are both used to find the derivative of a function, but they are used in different situations. The product rule is used when you need to find the derivative of a function that is the product of two or more functions, while the chain rule is used when you need to find the derivative of a function that is the composition of two or more functions. For example, if we have a function of the form $f(x) = (x^2 + 1)^3$ , then we would use the chain rule to find the derivative of $f(x)$ , while if we have a function of the form $f(x) = (x + 1)(x - 1)$ , then we would use the product rule to find the derivative of $f(x)$ .

Q: Can I use the product rule with trigonometric functions?

A: Yes, you can use the product rule with trigonometric functions. For example, if we have a function of the form $f(x) = \sin(x)\cos(x)$ , then we can use the product rule to find the derivative of $f(x)$ :

f'(x) = \cos(x)\cos(x) + \sin(x)(-\sin(x))

f'(x) = \cos^2(x) - \sin^2(x)

Conclusion

In this article, we have answered some of the most frequently asked questions about the product rule, a fundamental concept in calculus. We have discussed when to use the product rule, how to apply it, and some common mistakes to avoid. We have also discussed how to use the product rule with more than two factors, and how to know when to use the product rule versus the chain rule.