How can I use the power and quotient rule to solve this problem?

In summary, the conversation is about a math problem involving x/sqrt(x^2+1). The person has tried solving it using both the quotient and power rules but is getting different answers. They share their results and ask for clarification on how to combine fractions.
  • #1
xvtsx
15
0
Hi everyone,
I have been trying to do this problem in both ways but I can't get the same answer the book says. This is the problem:

x/ sqrt (x^2 +1)

With quotient rule I got until the point I have [(x^2 +1)^1/2 - x^2/(x^2 +1)^1/2]/(x^2 +1)
And with power rule I have [1/sqrt(x^2 +1)] - [x^2/(x^2 +1)^3/2]

If you guys can walk me through the problem, it would be nice. Thanks :wink:
 
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  • #2
Well, let's look at your result from the quotient rule:
[tex]\frac{\sqrt{x^{2}+1}-\frac{x^{2}}{\sqrt{x^{2}+1}}}{x^{2}+1}=\frac{\sqrt{x^{2}+1}}{x^{2}+1}-\frac{x^{2}}{(x^{2}+1)\sqrt{x^{2}+1}}=\frac{1}{\sqrt{x^{2}+1}}-\frac{x^{2}}{(x^{2}+1)^{\frac{3}{2}}}[/tex]

Does that look familiar?

Furthermore, we may expand our first fraction with the factor (x^{2}+1).
Then, we get:

[tex]\frac{(x^{2}+1)-x^{2}}{(x^{2}+1)^{\frac{3}{2}}}=\frac{1}{(x^{2}+1)^{\frac{3}{2}}}[/tex]
 
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  • #3
Can you explain me in details how you combined that part of the fraction because I get lost here [tex]\frac{\sqrt {x^{2}+1}}{x^{2}+1}-\frac{x^{2}}{(x^{2}+1)\sqrt{x^{2}+1}}=\frac{1}{\sq rt{x^{2}+1}}-\frac{x^{2}}{(x^{2}+1)^{\frac{3}{2}}}[/tex].
 
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  • #4
[tex]\frac{\sqrt {x^{2}+1}}{x^{2}+1}-\frac{x^{2}}{(x^{2}+1)\sqrt{x^{2}+1}}=\frac{1}{\sq rt{x^{2}+1}}-\frac{x^{2}}{(x^{2}+1)^{\frac{3}{2}}}[/tex]

You first multiply so all terms have a common divisor

[tex]\frac{\sqrt {x^{2}+1}*\sqrt{x^{2}+1}}{x^{2}+1}-\frac{x^{2}}{(x^{2}+1)\sqrt{x^{2}+1}}=\frac{1}{\sq rt{x^{2}+1}}-\frac{x^{2}}{(x^{2}+1)^{\frac{3}{2}}}[/tex]

and then simplify

as [tex]x^{2}+1[/tex] is the same as [tex](x^{2}+1)^{\frac{2}{2}[/tex] you multiply that with[tex](x^{2}+1)^{\frac{1}{2}[/tex]

Hope this helps
 

1. What is the Power Rule?

The Power Rule, also known as the Exponent Rule, is a mathematical rule used to find the derivative of a function that contains a variable raised to a power. It states that the derivative of x^n is nx^(n-1), where n is the exponent.

2. How is the Power Rule applied in calculus?

The Power Rule is used to find the derivative of a function, which is the slope of the function at any given point. This helps in solving problems related to rates of change and optimization in calculus.

3. What is the Quotient Rule?

The Quotient Rule is another mathematical rule used to find the derivative of a function. It is used when the function is in the form of f(x)/g(x), where f(x) and g(x) are both functions of x. The rule states that the derivative of f(x)/g(x) is (g(x)*f'(x)-f(x)*g'(x))/(g(x))^2.

4. When should the Quotient Rule be used instead of the Power Rule?

The Quotient Rule should be used when the function is a quotient of two functions, whereas the Power Rule should be used when the function is in the form of x^n. If the function is a combination of both, both rules may need to be used in combination.

5. Can the Power Rule and Quotient Rule be used together?

Yes, the Power Rule and Quotient Rule can be used together in certain cases. For example, if the function is in the form of (x^n)/g(x), both rules would need to be used to find the derivative. In other cases, only one rule may be applicable depending on the form of the function.

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