The Importance of the Power Rule

Oct 13, 2025 by miodeya.com 33 views

If $f(x)=(2x+7)^{-4}$ , find $f^{\prime}(x)$ and Then, find $f^{\prime}(5)$

In this article, we will explore the concept of finding the derivative of a function, specifically the function $f(x)=(2x+7)^{-4}$ . We will use the power rule of differentiation to find the derivative of this function and then evaluate it at a specific point, $x=5$ . This will help us understand how to apply the power rule in different scenarios and how to find the derivative of a function that involves a negative exponent.

The power rule of differentiation states that if $f(x)=x^n$ , then $f^{\prime}(x)=nx^{n-1}$ . This rule can be applied to functions that involve a variable raised to a power. In the case of the function $f(x)=(2x+7)^{-4}$ , we can use the power rule to find its derivative.

Finding the Derivative of $f(x)$

To find the derivative of $f(x)$ , we can use the chain rule of differentiation, which states that if $f(x)=g(h(x))$ , then $f^{\prime}(x)=g^{\prime}(h(x))\cdot h^{\prime}(x)$ . In this case, we have $f(x)=(2x+7)^{-4}$ , which can be written as $f(x)=g(h(x))$ , where $g(x)=x^{-4}$ and $h(x)=2x+7$ .

Using the chain rule, we can find the derivative of $f(x)$ as follows:

f^{\prime}(x)=g^{\prime}(h(x))\cdot h^{\prime}(x)

f^{\prime}(x)=(-4)(2x+7)^{-5}\cdot 2

f^{\prime}(x)=-8(2x+7)^{-5}

Evaluating the Derivative at $x=5$

Now that we have found the derivative of $f(x)$ , we can evaluate it at $x=5$ .

f^{\prime}(5)=-8(2(5)+7)^{-5}

f^{\prime}(5)=-8(17)^{-5}

f^{\prime}(5)=-8\cdot \frac{1}{17^5}

f^{\prime}(5)=-\frac{8}{247609}

In this article, we used the power rule of differentiation to find the derivative of the function $f(x)=(2x+7)^{-4}$ . We then evaluated the derivative at $x=5$ to find the value of $f^{\prime}(5)$ . This example illustrates how to apply the power rule in different scenarios and how to find the derivative of a function that involves a negative exponent.

The Importance of the Power Rule

The power rule of differentiation is a fundamental concept in calculus that allows us to find the derivative of a function that involves a variable raised to a power. This rule is essential in many areas of mathematics and science, including physics, engineering, and economics.

Real-World Applications of the Power Rule

The power rule of differentiation has many real-world applications, including:

Physics: The power rule is used to find the velocity and acceleration of an object in terms of its position and time.
Engineering: The power rule is used to find the derivative of a function that represents the stress or strain on a material.
Economics: The power rule is used to find the derivative of a function that represents the demand or supply of a product.

Common Mistakes to Avoid

When using the power rule of differentiation, there are several common mistakes to avoid, including:

Forgetting to apply the chain rule: When differentiating a function that involves a composite function, it is essential to apply the chain rule.
Not simplifying the derivative: After finding the derivative, it is essential to simplify it to its simplest form.
Not checking the domain: Before evaluating the derivative, it is essential to check the domain of the function.

In our previous article, we explored the concept of the power rule of differentiation and how to apply it to find the derivative of a function that involves a variable raised to a power. In this article, we will answer some common questions related to the power rule of differentiation.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a fundamental concept in calculus that states that if $f(x)=x^n$ , then $f^{\prime}(x)=nx^{n-1}$ . This rule can be applied to functions that involve a variable raised to a power.

Q: How do I apply the power rule of differentiation?

A: To apply the power rule of differentiation, you need to follow these steps:

Identify the function that you want to differentiate.
Determine the power to which the variable is raised.
Apply the power rule by multiplying the power by the variable raised to the power minus one.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation is a fundamental concept in calculus that states that if $f(x)=g(h(x))$ , then $f^{\prime}(x)=g^{\prime}(h(x))\cdot h^{\prime}(x)$ . This rule is used to find the derivative of a composite function.

Q: How do I apply the chain rule of differentiation?

A: To apply the chain rule of differentiation, you need to follow these steps:

Identify the composite function that you want to differentiate.
Determine the outer function and the inner function.
Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function.

Q: What is the difference between the power rule and the chain rule?

A: The power rule of differentiation is used to find the derivative of a function that involves a variable raised to a power, while the chain rule of differentiation is used to find the derivative of a composite function.

Q: Can I use the power rule and the chain rule together?

A: Yes, you can use the power rule and the chain rule together to find the derivative of a function that involves a composite function.

Q: What are some common mistakes to avoid when using the power rule and the chain rule?

A: Some common mistakes to avoid when using the power rule and the chain rule include:

Forgetting to apply the chain rule when differentiating a composite function.
Not simplifying the derivative after finding it.
Not checking the domain of the function before evaluating the derivative.

Q: How do I evaluate the derivative of a function at a specific point?

A: To evaluate the derivative of a function at a specific point, you need to follow these steps:

Find the derivative of the function.
Substitute the specific point into the derivative.
Simplify the expression to find the value of the derivative at that point.

Q: What are some real-world applications of the power rule and the chain rule?

A: The power rule and the chain rule have many real-world applications, including:

Physics: The power rule and the chain rule are used to find the velocity and acceleration of an object in terms of its position and time.
Engineering: The power rule and the chain rule are used to find the derivative of a function that represents the stress or strain on a material.
Economics: The power rule and the chain rule are used to find the derivative of a function that represents the demand or supply of a product.

In conclusion, the power rule and the chain rule are fundamental concepts in calculus that allow us to find the derivative of a function that involves a variable raised to a power or a composite function. By understanding these rules and their applications, we can solve a wide range of problems in mathematics and science.