# Trouble finding the derivative of a fraction using four step process

• MHB
• cole03k
In summary, the conversation involves trying to find the derivative of a function using the four step process, but encountering difficulties with the third step of dealing with fractions. The work provided shows the correct steps for finding the derivative and emphasizes the importance of understanding arithmetic and algebra before learning calculus.
cole03k

I am trying to find the derivative of this problem using the four step process but keep getting stuck when it comes to the third step of f(x+h) - f(x). I do not know what to do once I reach that step. Am I canceling terms out incorrectly? How should I deal with a fraction over a fraction? Any help would be really appreciated.

Work:

Note: Don't take short cuts with the fractions. $$\displaystyle \dfrac{1}{x - 6 - h} - \dfrac{1}{x - 6}$$ is not $$\displaystyle \dfrac{1}{(x - 6 - h) - (x - 6)}$$. No matter what your shorthand is you still have to get a common denominator and subtract the two fractions.

$$\displaystyle f(x) = \dfrac{1}{x - 6}$$

$$\displaystyle f'(x) = \lim_{h \to 0} \dfrac{ \dfrac{1}{x - 6 + h} - \dfrac{1}{x - 6} }{h}$$

$$\displaystyle f'(x) = \lim_{h \to 0} \dfrac{ \dfrac{(x - 6) - (x - 6 + h)}{(x - 6)(x - 6 + h)} }{h}$$

$$\displaystyle f'(x) = \lim_{h \ to 0} \dfrac{ \dfrac{-h}{(x - 6)(x - 6 + h)} }{h}$$

$$\displaystyle f'(x) = \lim_{h \to 0} \dfrac{-1}{(x - 6)(x - 6 + h)}$$

$$\displaystyle f'(x) = - \dfrac{1}{(x - 6)^2}$$

-Dan

The problem appears to be that you have never actually learned arithmetic!
First, $-\frac{a}{b}$ is NOT equal to $\frac{-a}{-b}$.
Second, $\frac{1}{a}- \frac{1}{b}$ is NOT equal to $\frac{1}{a- b}$..

Learn arithmetic and algebra before you attempt to learn Calculus!

## 1. How do I find the derivative of a fraction using the four step process?

The four step process for finding the derivative of a fraction involves the following steps:

1. Identify the numerator and denominator of the fraction
2. Apply the power rule to the numerator
3. Apply the quotient rule to the fraction
4. Simplify the resulting expression

## 2. Can I use the four step process to find the derivative of any fraction?

Yes, the four step process can be used to find the derivative of any fraction, as long as the fraction is in the form of a quotient of two functions.

## 3. What is the power rule and how is it applied in finding the derivative of a fraction?

The power rule states that the derivative of a function raised to a constant power is equal to the constant power multiplied by the derivative of the function. In the four step process, the power rule is applied to the numerator of the fraction.

## 4. How is the quotient rule applied in finding the derivative of a fraction?

The quotient rule states that the derivative of a quotient of two functions is equal to the denominator multiplied by the derivative of the numerator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator. In the four step process, the quotient rule is applied to the fraction after the power rule has been applied to the numerator.

## 5. Are there any shortcuts or alternative methods for finding the derivative of a fraction?

Yes, there are other methods for finding the derivative of a fraction, such as using the product rule or the chain rule. However, the four step process is a simple and straightforward method that can be used for any fraction. It is recommended to practice and understand the four steps before moving on to other methods.

• Calculus
Replies
8
Views
653
• Calculus
Replies
5
Views
2K
• Calculus
Replies
3
Views
3K
• Calculus
Replies
3
Views
1K
• Calculus
Replies
4
Views
1K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
6
Views
1K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
11
Views
2K
• Calculus
Replies
5
Views
1K