Calculus Problem: Find Derivative of f(x)= (x+1)/(x+2)(3x^2 + 6x)

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In summary, to find the derivative of f(x) = \frac{x+1}{x+2} (3x^2 + 6x), you can use either the product rule or the quotient rule. It is your choice which one you use first. Alternatively, you can simplify the expression before taking the derivative to avoid using the rules.
  • #1
Taturana
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Homework Statement



Find the derivative of:

[tex]f(x) = \frac{x+1}{x+2} (3x^2 + 6x)[/tex]

Homework Equations


The Attempt at a Solution



I tried but I don't know what rules should I apply here, so it's a waste of database space post here my wrong solution...

(Should I first to the derivative of the fraction and then to the product rule using the derivative of the fraction and the other thing in the parenthesis?)

Thank you,
Rafael Andreatta
 
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  • #2
Use the product rule and then the quotient rule.

[tex]
f(x) = \frac{x+1}{x+2} (3x^2 + 6x)
[/tex]


u=(x+1)/(x+2) du/dx = quotient rule.

v=3x2 + 6x
 
  • #3
rock.freak667 said:
Use the product rule and then the quotient rule.

[tex]
f(x) = \frac{x+1}{x+2} (3x^2 + 6x)
[/tex]


u=(x+1)/(x+2) du/dx = quotient rule.

v=3x2 + 6x

So I first use the product rule for [tex](x+1)(3x^2 + 6x)[/tex] and then the quotient rule between the result of the product rule and the (x+2)?

Why can't I do first eh quotient rule and then the product rule?
 
  • #4
You can, it's your choice what you use. I might write:
[tex]
\frac{x+1}{x+2} (3x^2 + 6x)=\frac{3x(x+1)(x+2)}{x+2}=3x^{2}+3x
[/tex]
Then I don't need to apply the product rule or the quotient rule.
 
  • #5
Taturana said:
So I first use the product rule for [tex](x+1)(3x^2 + 6x)[/tex] and then the quotient rule between the result of the product rule and the (x+2)?

Why can't I do first eh quotient rule and then the product rule?

You can do it however you wish!

hunt_mat said:
You can, it's your choice what you use. I might write:
[tex]
\frac{x+1}{x+2} (3x^2 + 6x)=\frac{3x(x+1)(x+2)}{x+2}=3x^{2}+3x
[/tex]
Then I don't need to apply the product rule or the quotient rule.

Like hunt_mat shows you.
 
  • #6
Ok, thank you all
 

1. What is the purpose of finding the derivative of a function?

The derivative of a function represents the rate of change of the function at a given point. It can also be thought of as the slope of the tangent line to the function at that point. Finding the derivative allows us to understand the behavior of the function and make predictions about its future values.

2. How do you find the derivative of a function using the quotient rule?

The quotient rule states that the derivative of a function, f(x), divided by another function, g(x), is equal to (g(x)*f'(x) - f(x)*g'(x)) / (g(x))^2. In the case of f(x)= (x+1)/(x+2)(3x^2 + 6x), we would apply the quotient rule to the numerator and denominator separately and then simplify the resulting expression.

3. Can you find the derivative of a function using other rules or methods?

Yes, there are other methods for finding derivatives such as the product rule, chain rule, and power rule. In some cases, it may be more efficient to use one of these rules instead of the quotient rule.

4. What is the significance of the derivative being equal to 0?

If the derivative of a function is equal to 0 at a certain point, it means that the function is not changing at that point. This can be useful in finding maximum and minimum values of a function, as these occur when the derivative is equal to 0.

5. How can the derivative of a function be used in real-life applications?

The derivative has many real-life applications, such as in physics, economics, and engineering. For example, in physics, the derivative of an object's position with respect to time gives its velocity, and the derivative of velocity gives its acceleration. In economics, the derivative can be used to find marginal cost and marginal revenue. In engineering, the derivative can be used to optimize designs and predict the behavior of systems.

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