Find The Derivative Of The Function:${ \frac{dy}{dx} = \frac{4}{x} + \sqrt[4]{x^3} - \cos X }$

Mar 11, 2025 by ADMIN 96 views

**Derivative of a Function: A Comprehensive Guide**

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will discuss the derivative of a given function and provide a step-by-step guide on how to find it.

The Given Function

The given function is:

\frac{dy}{dx} = \frac{4}{x} + \sqrt[4]{x^3} - \cos x

This function consists of three terms: a rational term, a radical term, and a trigonometric term. To find the derivative of this function, we will apply the power rule, the chain rule, and the sum rule of differentiation.

Applying the Power Rule

The power rule states that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . We will apply this rule to the rational term and the radical term.

Rational Term

The rational term is $\frac{4}{x}$ . Using the power rule, we can rewrite this term as $4x^{-1}$ . The derivative of this term is:

\frac{d}{dx} \left( 4x^{-1} \right) = -4x^{-2} = -\frac{4}{x^2}

Radical Term

The radical term is $\sqrt[4]{x^3}$ . Using the power rule, we can rewrite this term as $x^{\frac{3}{4}}$ . The derivative of this term is:

\frac{d}{dx} \left( x^{\frac{3}{4}} \right) = \frac{3}{4}x^{\frac{3}{4}-1} = \frac{3}{4}x^{-\frac{1}{4}} = \frac{3}{4\sqrt[4]{x}}

Applying the Chain Rule

The chain rule states that if $f(x) = g(h(x))$ , then $f'(x) = g'(h(x)) \cdot h'(x)$ . We will apply this rule to the trigonometric term.

Trigonometric Term

The trigonometric term is $-\cos x$ . Using the chain rule, we can rewrite this term as $-\cos(x)$ . The derivative of this term is:

\frac{d}{dx} \left( -\cos(x) \right) = \sin(x)

Combining the Terms

Now that we have found the derivatives of each term, we can combine them to find the derivative of the given function.

\frac{dy}{dx} = -\frac{4}{x^2} + \frac{3}{4\sqrt[4]{x}} - \sin(x)

Conclusion

In this article, we discussed the derivative of a given function and provided a step-by-step guide on how to find it. We applied the power rule, the chain rule, and the sum rule of differentiation to find the derivative of each term and then combined them to find the derivative of the given function. The derivative of the given function is:

\frac{dy}{dx} = -\frac{4}{x^2} + \frac{3}{4\sqrt[4]{x}} - \sin(x)

This derivative represents the rate of change of the given function with respect to $x$ . It has numerous applications in various fields, including physics, engineering, and economics.

Applications of Derivatives

Derivatives have numerous applications in various fields, including:

Physics: Derivatives are used to describe the motion of objects, including velocity, acceleration, and force.
Engineering: Derivatives are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
Economics: Derivatives are used to model economic systems, including supply and demand, inflation, and interest rates.
Computer Science: Derivatives are used in machine learning, optimization, and data analysis.

Limitations of Derivatives

While derivatives are a powerful tool for modeling and analyzing functions, they have some limitations. Some of these limitations include:

Differentiability: Derivatives require the function to be differentiable at a point. If the function is not differentiable, the derivative may not exist.
Domain: Derivatives are only defined for functions that are defined on an interval. If the function is not defined on an interval, the derivative may not exist.
Computational complexity: Derivatives can be computationally intensive, especially for complex functions.

Conclusion

In conclusion, derivatives are a fundamental concept in mathematics and have numerous applications in various fields. They are used to describe the rate of change of a function with respect to one of its variables. While derivatives have some limitations, they are a powerful tool for modeling and analyzing functions. In this article, we discussed the derivative of a given function and provided a step-by-step guide on how to find it. We applied the power rule, the chain rule, and the sum rule of differentiation to find the derivative of each term and then combined them to find the derivative of the given function.

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Q&A: Derivative of a Function

Q: What is the derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to one of its variables. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you can use the power rule, the chain rule, and the sum rule of differentiation. The power rule states that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . The chain rule states that if $f(x) = g(h(x))$ , then $f'(x) = g'(h(x)) \cdot h'(x)$ . The sum rule states that if $f(x) = g(x) + h(x)$ , then $f'(x) = g'(x) + h'(x)$ .

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . This means that if you have a function of the form $x^n$ , you can find its derivative by multiplying the exponent by the coefficient and then subtracting 1 from the exponent.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation states that if $f(x) = g(h(x))$ , then $f'(x) = g'(h(x)) \cdot h'(x)$ . This means that if you have a function of the form $g(h(x))$ , you can find its derivative by first finding the derivative of the inner function $h(x)$ and then multiplying it by the derivative of the outer function $g(x)$ .

Q: What is the sum rule of differentiation?

A: The sum rule of differentiation states that if $f(x) = g(x) + h(x)$ , then $f'(x) = g'(x) + h'(x)$ . This means that if you have a function of the form $g(x) + h(x)$ , you can find its derivative by finding the derivatives of the individual functions $g(x)$ and $h(x)$ and then adding them together.

Q: How do I apply the power rule, chain rule, and sum rule of differentiation?

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

Identify the type of function you are working with (e.g. polynomial, trigonometric, exponential).
Apply the appropriate rule of differentiation (e.g. power rule, chain rule, sum rule).
Simplify the resulting expression to find the derivative.

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, chain rule, or sum rule of differentiation.
Making errors in the application of the rules (e.g. forgetting to multiply by the exponent or forgetting to add the derivatives).
Not simplifying the resulting expression to find the derivative.

Q: How do I check my work when finding the derivative of a function?

A: To check your work when finding the derivative of a function, you can use the following steps:

Plug the derivative back into the original function to see if it is true.
Check the units of the derivative to make sure they are correct.
Check the sign of the derivative to make sure it is correct.

Q: What are some real-world applications of derivatives?

A: Derivatives have numerous real-world applications, including:

Physics: Derivatives are used to describe the motion of objects, including velocity, acceleration, and force.
Engineering: Derivatives are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
Economics: Derivatives are used to model economic systems, including supply and demand, inflation, and interest rates.
Computer Science: Derivatives are used in machine learning, optimization, and data analysis.

Q: What are some limitations of derivatives?

A: Some limitations of derivatives include:

Differentiability: Derivatives require the function to be differentiable at a point. If the function is not differentiable, the derivative may not exist.
Domain: Derivatives are only defined for functions that are defined on an interval. If the function is not defined on an interval, the derivative may not exist.
Computational complexity: Derivatives can be computationally intensive, especially for complex functions.

Q: How do I find the derivative of a function with multiple variables?

A: To find the derivative of a function with multiple variables, you can use the following steps:

Identify the variables in the function.
Apply the chain rule of differentiation to find the derivative of each variable.
Combine the derivatives of each variable to find the derivative of the function.

Q: What are some common functions that have derivatives?

A: Some common functions that have derivatives include:

Polynomials: $f(x) = x^n$
Trigonometric functions: $f(x) = \sin(x)$ , $f(x) = \cos(x)$
Exponential functions: $f(x) = e^x$
Logarithmic functions: $f(x) = \log(x)$

Q: How do I find the derivative of a function with a constant multiple?

A: To find the derivative of a function with a constant multiple, you can use the following steps:

Identify the constant multiple.
Apply the power rule of differentiation to find the derivative of the function.
Multiply the derivative by the constant multiple.

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

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

Forgetting to multiply the derivative by the constant multiple.
Making errors in the application of the power rule of differentiation.
Not simplifying the resulting expression to find the derivative.