Let $f(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}$.Find $f^{\prime}(x$\]. $\[ \square \\](Note: Question Help: Video, Add Work Sections Are Not Part Of The Main Question.)

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Introduction

In this article, we will explore the concept of differentiation and apply it to a complex function. The function we will be working with is f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}. Our goal is to find the derivative of this function, denoted as f′(x)f^{\prime}(x).

What is Differentiation?

Differentiation is a fundamental concept in calculus that deals with the study of rates of change and slopes of curves. It is a mathematical operation that takes a function and produces another function that represents the rate of change of the original function. In other words, differentiation helps us understand how a function changes as its input changes.

The Power Rule and the Chain Rule

To find the derivative of the given function, we will use two important rules in differentiation: the power rule and the chain rule.

  • Power Rule: If f(x)=xnf(x) = x^n, then f′(x)=nxn−1f^{\prime}(x) = nx^{n-1}.
  • Chain Rule: If f(x)=g(h(x))f(x) = g(h(x)), then f′(x)=g′(h(x))â‹…h′(x)f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x).

Finding the Derivative of the Given Function

To find the derivative of the given function, we will apply the power rule and the chain rule.

Step 1: Simplify the Function

First, let's simplify the given function by combining the two terms.

f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}

We can rewrite the function as:

f(x)=4x4â‹…x12+5x3â‹…x12f(x) = 4x^4 \cdot x^{\frac{1}{2}} + \frac{5}{x^3 \cdot x^{\frac{1}{2}}}

Using the properties of exponents, we can simplify the function further:

f(x)=4x92+5x72f(x) = 4x^{\frac{9}{2}} + \frac{5}{x^{\frac{7}{2}}}

Step 2: Apply the Power Rule

Now, let's apply the power rule to each term in the function.

For the first term, we have:

f′(x)=ddx(4x92)f^{\prime}(x) = \frac{d}{dx} (4x^{\frac{9}{2}})

Using the power rule, we get:

f′(x)=4⋅92x92−1f^{\prime}(x) = 4 \cdot \frac{9}{2} x^{\frac{9}{2} - 1}

Simplifying further, we get:

f′(x)=18x52f^{\prime}(x) = 18x^{\frac{5}{2}}

For the second term, we have:

f′(x)=ddx(5x72)f^{\prime}(x) = \frac{d}{dx} \left(\frac{5}{x^{\frac{7}{2}}}\right)

Using the power rule, we get:

f′(x)=−52x−72−1f^{\prime}(x) = -\frac{5}{2} x^{-\frac{7}{2} - 1}

Simplifying further, we get:

f′(x)=−52x−92f^{\prime}(x) = -\frac{5}{2} x^{-\frac{9}{2}}

Step 3: Combine the Results

Now, let's combine the results from the previous steps.

f′(x)=18x52−52x−92f^{\prime}(x) = 18x^{\frac{5}{2}} - \frac{5}{2} x^{-\frac{9}{2}}

Conclusion

In this article, we found the derivative of the complex function f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}. We applied the power rule and the chain rule to simplify the function and find its derivative. The final result is:

f′(x)=18x52−52x−92f^{\prime}(x) = 18x^{\frac{5}{2}} - \frac{5}{2} x^{-\frac{9}{2}}

Introduction

In our previous article, we explored the concept of differentiation and applied it to a complex function. We found the derivative of the function f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}. In this article, we will answer some frequently asked questions related to the derivative of this function.

Q: What is the derivative of the function f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}?

A: The derivative of the function f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}} is f′(x)=18x52−52x−92f^{\prime}(x) = 18x^{\frac{5}{2}} - \frac{5}{2} x^{-\frac{9}{2}}.

Q: How did you find the derivative of the function?

A: We found the derivative of the function by applying the power rule and the chain rule. We first simplified the function by combining the two terms, and then applied the power rule to each term.

Q: What is the power rule?

A: The power rule is a rule in differentiation that states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f^{\prime}(x) = nx^{n-1}.

Q: What is the chain rule?

A: The chain rule is a rule in differentiation that states that if f(x)=g(h(x))f(x) = g(h(x)), then f′(x)=g′(h(x))⋅h′(x)f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x).

Q: How do you apply the power rule and the chain rule?

A: To apply the power rule, you simply multiply the coefficient of the term by the exponent and then subtract 1 from the exponent. To apply the chain rule, you first find the derivative of the outer function and then multiply it by the derivative of the inner function.

Q: What is the significance of the derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to its input. It is a measure of how fast the function changes as its input changes.

Q: How do you use the derivative of a function in real-world applications?

A: The derivative of a function is used in a variety of real-world applications, including physics, engineering, and economics. For example, the derivative of a function can be used to model the motion of an object, the growth of a population, or the behavior of a financial market.

Q: What are some common mistakes to avoid when finding the derivative of a function?

A: Some common mistakes to avoid when finding the derivative of a function include:

  • Forgetting to apply the power rule or the chain rule
  • Making errors in the application of the power rule or the chain rule
  • Failing to simplify the function before finding its derivative
  • Not checking the units of the derivative

Conclusion

In this article, we answered some frequently asked questions related to the derivative of the complex function f(x)=4x4x+5x3xf(x) = 4x^4 \sqrt{x} + \frac{5}{x^3 \sqrt{x}}. We provided explanations and examples to help clarify the concepts and procedures involved in finding the derivative of a function.