Find The Derivative Of The Function:${ Y = \frac{e {11x}}{x 7} }$ {\frac{dy}{dx} = \}

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Introduction

In this article, we will explore the concept of finding the derivative of a function, specifically the function $y = \frac{e^{11x}}{x^7}$. The derivative of a function represents the rate of change of the function with respect to the variable. It is a fundamental concept in calculus and has numerous applications in various fields such as physics, engineering, and economics.

What is the Derivative?

The derivative of a function $f(x)$ is denoted as $f'(x)$ and represents the rate of change of the function with respect to the variable $x$. It is calculated by taking the limit of the difference quotient as the change in $x$ approaches zero.

The Quotient Rule

To find the derivative of the function $y = \frac{e^{11x}}{x^7}$, we will use the quotient rule, which states that if $y = \frac{f(x)}{g(x)}$, then $y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.

Applying the Quotient Rule

Let's apply the quotient rule to the function $y = \frac{e^{11x}}{x^7}$. We have:

$f(x) = e^{11x}$ and $g(x) = x^7$

Finding the Derivatives of $f(x)$ and $g(x)$

To apply the quotient rule, we need to find the derivatives of $f(x)$ and $g(x)$.

The derivative of $f(x) = e^{11x}$ is $f'(x) = 11e^{11x}$.

The derivative of $g(x) = x^7$ is $g'(x) = 7x^6$.

Applying the Quotient Rule

Now that we have the derivatives of $f(x)$ and $g(x)$, we can apply the quotient rule to find the derivative of $y = \frac{e^{11x}}{x^7}$.

$y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

$y' = \frac{(11e^{11x})(x^7) - (e^{11x})(7x^6)}{(x^7)^2}$

$y' = \frac{11e^{11x}x^7 - 7e^{11x}x^6}{x^{14}}$

Simplifying the Derivative

We can simplify the derivative by factoring out the common term $e^{11x}$.

$y' = e^{11x} \frac{11x^7 - 7x^6}{x^{14}}$

$y' = e^{11x} \frac{x^6(11x - 7)}{x^{14}}$

$y' = e^{11x} \frac{11x - 7}{x^7}$

Conclusion

In this article, we found the derivative of the function $y = \frac{e^{11x}}{x^7}$ using the quotient rule. The derivative is given by $y' = e^{11x} \frac{11x - 7}{x^7}$. This result can be used to analyze the behavior of the function and its rate of change with respect to the variable $x$.

Applications of the Derivative

The derivative of the function $y = \frac{e^{11x}}{x^7}$ has numerous applications in various fields such as physics, engineering, and economics. For example, it can be used to model the growth and decay of populations, the motion of objects, and the behavior of economic systems.

Final Thoughts

In conclusion, finding the derivative of a function is a fundamental concept in calculus that has numerous applications in various fields. The quotient rule is a powerful tool for finding the derivative of a function, and it can be used to analyze the behavior of the function and its rate of change with respect to the variable.

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Introduction

In our previous article, we explored the concept of finding the derivative of a function, specifically the function $y = \frac{e^{11x}}{x^7}$. We used the quotient rule to find the derivative, which is given by $y' = e^{11x} \frac{11x - 7}{x^7}$. In this article, we will answer some frequently asked questions about finding the derivative of this function.

Q: What is the quotient rule?

A: The quotient rule is a formula for finding the derivative of a function that is a quotient of two functions. It is given by $y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$, where $f(x)$ and $g(x)$ are the two functions.

Q: How do I apply the quotient rule to find the derivative of a function?

A: To apply the quotient rule, you need to find the derivatives of the two functions $f(x)$ and $g(x)$, and then plug them into the formula $y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.

Q: What is the derivative of $e^{11x}$?

A: The derivative of $e^{11x}$ is $11e^{11x}$.

Q: What is the derivative of $x^7$?

A: The derivative of $x^7$ is $7x^6$.

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

A: To simplify the derivative of a function, you can try to factor out common terms, cancel out any common factors, and rewrite the expression in a simpler form.

Q: What are some common applications of the derivative of a function?

A: The derivative of a function has numerous applications in various fields such as physics, engineering, and economics. Some common applications include modeling the growth and decay of populations, the motion of objects, and the behavior of economic systems.

Q: How do I use the derivative of a function to analyze the behavior of the function?

A: To analyze the behavior of a function, you can use the derivative to determine the rate of change of the function with respect to the variable. You can also use the derivative to find the maximum and minimum values of the function, and to determine the intervals where the function is increasing or decreasing.

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 quotient rule when the function is a quotient of two functions
• Not simplifying the derivative of a function
• Not checking the domain of the function before finding the derivative
• Not using the correct formula for the derivative of a function

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 try the following:

• Plug the derivative back into the original function to see if it is true
• Check the domain of the function to make sure it is correct
• Simplify the derivative to see if it is in the correct form
• Use a calculator or computer program to check the derivative

Conclusion

In this article, we answered some frequently asked questions about finding the derivative of the function $y = \frac{e^{11x}}{x^7}$. We hope that this article has been helpful in clarifying any confusion about finding the derivative of a function. If you have any further questions, please don't hesitate to ask.