Find The Derivative Of The Function.$\[ Y = 4x^{-5} - 7x^{-1} \\]$\[\frac{dy}{dx} = \square \\]

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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. In this article, we will focus on finding the derivative of a given function, which is a crucial concept in mathematics and has numerous applications in various fields such as physics, engineering, and economics.

The Function

The given function is:

y=4x−5−7x−1{ y = 4x^{-5} - 7x^{-1} }

This function is a combination of two terms, each of which is a power function. The first term is 4x−54x^{-5}, and the second term is −7x−1-7x^{-1}.

The Derivative

To find the derivative of the function, we will use the power rule of differentiation, which states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}.

Step 1: Differentiate the First Term

The first term is 4x−54x^{-5}. Using the power rule, we can differentiate this term as follows:

ddx(4x−5)=4⋅(−5)x−5−1{ \frac{d}{dx} (4x^{-5}) = 4 \cdot (-5) x^{-5-1} } =−20x−6{ = -20x^{-6} }

Step 2: Differentiate the Second Term

The second term is −7x−1-7x^{-1}. Using the power rule, we can differentiate this term as follows:

ddx(−7x−1)=−7⋅(−1)x−1−1{ \frac{d}{dx} (-7x^{-1}) = -7 \cdot (-1) x^{-1-1} } =7x−2{ = 7x^{-2} }

Step 3: Combine the Results

Now that we have differentiated both terms, we can combine the results to find the derivative of the function:

dydx=−20x−6+7x−2{ \frac{dy}{dx} = -20x^{-6} + 7x^{-2} }

Simplifying the Derivative

We can simplify the derivative by combining the two terms:

dydx=−20x−6+7x−2{ \frac{dy}{dx} = -20x^{-6} + 7x^{-2} } =−20x6+7x2{ = \frac{-20}{x^6} + \frac{7}{x^2} }

Conclusion

In this article, we have found the derivative of a given function using the power rule of differentiation. The derivative of the function is:

dydx=−20x6+7x2{ \frac{dy}{dx} = \frac{-20}{x^6} + \frac{7}{x^2} }

This derivative represents the rate of change of the function with respect to the variable xx. Understanding the concept of derivatives is crucial in mathematics and has numerous applications in various fields.

Applications of Derivatives

Derivatives have numerous applications in various fields such as physics, engineering, and economics. Some of the applications of derivatives include:

  • Optimization: Derivatives are used to find the maximum or minimum value of a function.
  • Physics: Derivatives are used to describe the motion of objects and the forces acting on them.
  • Engineering: Derivatives are used to design and optimize systems such as bridges, buildings, and electronic circuits.
  • Economics: Derivatives are used to model the behavior of economic systems and make predictions about future trends.

Conclusion

In conclusion, finding the derivative of a function is a crucial concept in mathematics that has numerous applications in various fields. In this article, we have found the derivative of a given function using the power rule of differentiation. The derivative of the function is:

dydx=−20x6+7x2{ \frac{dy}{dx} = \frac{-20}{x^6} + \frac{7}{x^2} }

Introduction

In our previous article, we discussed finding the derivative of a function using the power rule of differentiation. In this article, we will answer some frequently asked questions about derivatives.

Q: What is a derivative?

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

Q: Why are derivatives important?

A: Derivatives are important because they help us understand how functions change and behave. They have numerous applications in various fields such as 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 of differentiation, which states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}. You can also use other rules such as the product rule and the quotient rule.

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}. This rule is used to find the derivative of a function that is a power function.

Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if f(x)=u(x)v(x)f(x) = u(x)v(x), then f′(x)=u′(x)v(x)+u(x)v′(x)f'(x) = u'(x)v(x) + u(x)v'(x). This rule is used to find the derivative of a function that is a product of two functions.

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation states that if f(x)=u(x)v(x)f(x) = \frac{u(x)}{v(x)}, then f′(x)=u′(x)v(x)−u(x)v′(x)v(x)2f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}. This rule is used to find the derivative of a function that is a quotient of two functions.

Q: How do I use the derivative to solve problems?

A: To use the derivative to solve problems, you can use it to find the rate of change of a function, the maximum or minimum value of a function, and the behavior of a function.

Q: What are some common applications of derivatives?

A: Some common applications of derivatives include:

  • Optimization: Derivatives are used to find the maximum or minimum value of a function.
  • Physics: Derivatives are used to describe the motion of objects and the forces acting on them.
  • Engineering: Derivatives are used to design and optimize systems such as bridges, buildings, and electronic circuits.
  • Economics: Derivatives are used to model the behavior of economic systems and make predictions about future trends.

Q: What are some common mistakes to avoid when finding derivatives?

A: Some common mistakes to avoid when finding derivatives include:

  • Forgetting to use the power rule: Make sure to use the power rule when finding the derivative of a power function.
  • Forgetting to use the product rule: Make sure to use the product rule when finding the derivative of a product of two functions.
  • Forgetting to use the quotient rule: Make sure to use the quotient rule when finding the derivative of a quotient of two functions.
  • Not checking the domain of the function: Make sure to check the domain of the function before finding its derivative.

Conclusion

In conclusion, derivatives are an important concept in mathematics that have numerous applications in various fields. In this article, we have answered some frequently asked questions about derivatives and provided some tips and tricks for finding derivatives. We hope this article has been helpful in understanding the concept of derivatives.