Find The Difference Quotient Of \[$ F \$\], That Is, Find \[$\frac{f(x+h)-f(x)}{h}, \, H \neq 0\$\], For The Following Function. Be Sure To Fully Simplify.Given:$\[ F(x) = \sqrt{17x} \\]Find:$\[ \frac{f(x+h)-f(x)}{h}

by ADMIN 217 views

Introduction

In calculus, the difference quotient is a fundamental concept used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero. In this article, we will find the difference quotient of the given function f(x)=17x{ f(x) = \sqrt{17x} } and fully simplify the result.

The Difference Quotient Formula

The difference quotient formula is given by:

f(x+h)−f(x)h, h≠0{ \frac{f(x+h)-f(x)}{h}, \, h \neq 0 }

This formula represents the average rate of change of the function over a small interval of length hh.

Finding the Difference Quotient of the Given Function

To find the difference quotient of the given function, we need to substitute the function into the difference quotient formula and simplify the result.

f(x+h)−f(x)h=17(x+h)−17xh{ \frac{f(x+h)-f(x)}{h} = \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} }

Simplifying the Expression

To simplify the expression, we can start by rationalizing the numerator.

17(x+h)−17xh=17(x+h)−17xh⋅17(x+h)+17x17(x+h)+17x{ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} \cdot \frac{\sqrt{17(x+h)} + \sqrt{17x}}{\sqrt{17(x+h)} + \sqrt{17x}} }

This will eliminate the square roots in the numerator.

Expanding and Simplifying

Expanding and simplifying the expression, we get:

17(x+h)−17xh=17(x+h)−17xh(17(x+h)+17x){ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{17(x+h) - 17x}{h(\sqrt{17(x+h)} + \sqrt{17x})} }

Simplifying further, we get:

17(x+h)−17xh=17hh(17(x+h)+17x){ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{17h}{h(\sqrt{17(x+h)} + \sqrt{17x})} }

Canceling Out the Common Factor

Canceling out the common factor of hh, we get:

17(x+h)−17xh=1717(x+h)+17x{ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{17}{\sqrt{17(x+h)} + \sqrt{17x}} }

Simplifying the Expression Further

Simplifying the expression further, we get:

17(x+h)−17xh=1717x(1+hx+1){ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{17}{\sqrt{17x}(\sqrt{1+\frac{h}{x}} + 1)} }

Final Simplification

Finally, simplifying the expression, we get:

17(x+h)−17xh=1717x(1+hx+1){ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{17}{17\sqrt{x}(\sqrt{1+\frac{h}{x}} + 1)} }

17(x+h)−17xh=1x(1+hx+1){ \frac{\sqrt{17(x+h)} - \sqrt{17x}}{h} = \frac{1}{\sqrt{x}(\sqrt{1+\frac{h}{x}} + 1)} }

Conclusion

In this article, we found the difference quotient of the given function f(x)=17x{ f(x) = \sqrt{17x} } and fully simplified the result. The difference quotient is a fundamental concept in calculus used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart

Discussion

The difference quotient is a fundamental concept in calculus used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero. In this article, we found the difference quotient of the given function f(x)=17x{ f(x) = \sqrt{17x} } and fully simplified the result.

Related Topics

  • Derivatives
  • Limits
  • Calculus

Keywords

  • Difference quotient
  • Derivative
  • Calculus
  • Limits
  • Rationalizing the numerator
  • Simplifying the expression
  • Canceling out the common factor
    Difference Quotient Q&A ==========================

Introduction

In our previous article, we found the difference quotient of the function f(x)=17x{ f(x) = \sqrt{17x} } and fully simplified the result. In this article, we will answer some frequently asked questions about the difference quotient.

Q: What is the difference quotient?

A: The difference quotient is a fundamental concept in calculus used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero.

Q: How do I find the difference quotient of a function?

A: To find the difference quotient of a function, you need to substitute the function into the difference quotient formula and simplify the result.

Q: What is the difference quotient formula?

A: The difference quotient formula is given by:

f(x+h)−f(x)h, h≠0{ \frac{f(x+h)-f(x)}{h}, \, h \neq 0 }

Q: Can I use the difference quotient formula to find the derivative of a function?

A: Yes, you can use the difference quotient formula to find the derivative of a function. However, you need to take the limit of the difference quotient as the change in the input variable approaches zero.

Q: How do I simplify the difference quotient expression?

A: To simplify the difference quotient expression, you can start by rationalizing the numerator. This will eliminate the square roots in the numerator.

Q: Can I cancel out the common factor in the difference quotient expression?

A: Yes, you can cancel out the common factor in the difference quotient expression. This will simplify the expression further.

Q: What is the final simplified form of the difference quotient expression?

A: The final simplified form of the difference quotient expression is:

1x(1+hx+1){ \frac{1}{\sqrt{x}(\sqrt{1+\frac{h}{x}} + 1)} }

Q: Can I use the difference quotient to find the derivative of a function at a specific point?

A: Yes, you can use the difference quotient to find the derivative of a function at a specific point. However, you need to take the limit of the difference quotient as the change in the input variable approaches zero.

Q: What are some common mistakes to avoid when finding the difference quotient?

A: Some common mistakes to avoid when finding the difference quotient include:

  • Not rationalizing the numerator
  • Not canceling out the common factor
  • Not taking the limit of the difference quotient as the change in the input variable approaches zero

Conclusion

In this article, we answered some frequently asked questions about the difference quotient. The difference quotient is a fundamental concept in calculus used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart

Discussion

The difference quotient is a fundamental concept in calculus used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero. In this article, we answered some frequently asked questions about the difference quotient.

Related Topics

  • Derivatives
  • Limits
  • Calculus

Keywords

  • Difference quotient
  • Derivative
  • Calculus
  • Limits
  • Rationalizing the numerator
  • Simplifying the expression
  • Canceling out the common factor