Derivative of a Fraction: Simplifying with Polynomial Division?

In summary, when finding the derivative of a quotient, the quotient rule should be used instead of simply taking the derivative of each individual term. Additionally, it is important to use correct notation when expressing derivatives, with d/dx indicating the goal of taking the derivative and f' representing the derivative of a function.
  • #1
caters
229
9

Homework Statement


if $$y = \frac{2x^5-3x^3+x^2}{x^3}$$ then $$\frac{dy}{dx} =$$

Homework Equations


if $$f(x) = x^n$$ then $$f'(x) = nx^{n-1}$$

The Attempt at a Solution


$$\frac{2x^5-3x^3+x^2}{x^3} = \frac{2x^5}{x^3} - \frac{3x^3}{x^3} + \frac{x^2}{x^3}$$
$$ f'(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{10x^4}{3x^2} - \frac{9x^2}{3x^2} + \frac{2x}{3x^2}$$
$$ f'(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{10x^4}{3x^2} - 3 + \frac{2x}{3x^2}$$

And now I am stuck as to how to simplify this. Should I have done the whole polynomial division before I took the derivative?
 
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  • #2
caters said:

Homework Statement


if $$y = \frac{2x^5-3x^3+x^2}{x^3}$$ then $$\frac{dy}{dx} =$$

Homework Equations


if $$f(x) = x^n$$ then $$f'(x) = nx^{n-1}$$

The Attempt at a Solution


$$\frac{2x^5-3x^3+x^2}{x^3} = \frac{2x^5}{x^3} - \frac{3x^3}{x^3} + \frac{x^2}{x^3}$$
$$ f'(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{10x^4}{3x^2} - \frac{9x^2}{3x^2} + \frac{2x}{3x^2}$$
$$ f'(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{10x^4}{3x^2} - 3 + \frac{2x}{3x^2}$$

And now I am stuck as to how to simplify this. Should I have done the whole polynomial division before I took the derivative?

The derivative of [f(x) / g (x)] ≠ f'(x) / g'(x).

If you have the expression 2x5/x3, how would you simplify that before taking the derivative?
 
  • #3
caters said:
$$\frac{2x^5-3x^3+x^2}{x^3} = \frac{2x^5}{x^3} - \frac{3x^3}{x^3} + \frac{x^2}{x^3}$$
This is fine.

caters said:
$$ f'(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{10x^4}{3x^2} - \frac{9x^2}{3x^2} + \frac{2x}{3x^2}$$
This is not!

Look up the quotient rule.

caters said:
And now I am stuck as to how to simplify this. Should I have done the whole polynomial division before I took the derivative?
Yes.
 
  • #4
I read a site with derivative shortcuts and it said this "if you have a polynomial with only 1 term in the denominator than you can separate it into individual fractions and take the derivative of each of those fractions separately.

If I were to divide each term by x3 than I would get 2x2 - 3 + 1/x
 
  • #5
caters said:
I read a site with derivative shortcuts and it said this "if you have a polynomial with only 1 term in the denominator than you can separate it into individual fractions and take the derivative of each of those fractions separately.

If I were to divide each term by x3 than I would get 2x2 - 3 + 1/x
Nothing wrong with that. Now you can take the derivative of each term separately.
 
  • #6
Doc Al said:
Look up the quotient rule.

I think the OP is just starting to learn derivatives and he only knows the power rule.

I think the purpose of this exercise was to show how the original expression could be split up and simplified before applying the power rule. Unfortunately, the OP did not simplify and thought (incorrectly) that the derivative of a quotient was the quotient of the derivatives.
 
  • #7
Also not mentioned by the other folks responding in this thread is the incorrect use of notation.
##f'(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{10x^4}{3x^2} - \frac{9x^2}{3x^2} + \frac{2x}{3x^2}##
This does NOT mean the derivative of the quantity in parentheses. It means to evaluate f' at ##\frac{2x^5-3x^3+x^2}{x^3}##. A better use of notation would be the following:
##\frac{d}{dx}(\frac{2x^5-3x^3+x^2}{x^3}) = \frac{d}{dx}(\frac{10x^4}{3x^2} - \frac{9x^2}{3x^2} + \frac{2x}{3x^2}) = ... ##
 
  • #8
But how is d/dx any different from f'? They are just 2 notations for the same thing, the first derivative. Same with 2 of these d/dx and f'' for second derivative and so on for theoretically infinitely many derivatives.
 
  • #9
caters said:
But how is d/dx any different from f'? They are just 2 notations for the same thing, the first derivative. Same with 2 of these d/dx and f'' for second derivative and so on for theoretically infinitely many derivatives.

Typically, the derivative is expressed as follows: for a function y = f(x), y' = f'(x) = df(x)/dx, where x is the independent variable. x does not represent, for example, another function, which is the distinction that Mark44 was trying to make.

If f(x) = x2, then writing f(x2) should mean (x2)2 to be consistent with functional notation.
 
  • #10
SteamKing said:
Unfortunately, the OP did not simplify and thought (incorrectly) that the derivative of a quotient was the quotient of the derivatives.
I agree, of course. (Which is why I suggested that he look up the https://www.math.hmc.edu/calculus/tutorials/quotient_rule/, so he knows why what he did was wrong.)
 
  • #11
caters said:
But how is d/dx any different from f'? They are just 2 notations for the same thing, the first derivative. Same with 2 of these d/dx and f'' for second derivative and so on for theoretically infinitely many derivatives.
No, they are not the same. In addition to what SteamKing said, the operator d/dx indicates that the goal is to take the derivative of what follows it. The function f' is the derivative of some function f. It does not mean to take the derivative of what is in the parentheses.
 

Related to Derivative of a Fraction: Simplifying with Polynomial Division?

1. What is the derivative of a fraction?

The derivative of a fraction is the rate of change of the fraction with respect to its variable. It represents how much the fraction is changing for a given change in the variable.

2. How do you find the derivative of a fraction?

To find the derivative of a fraction, you can use the quotient rule, which states that the derivative of a fraction is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

3. Can you simplify the derivative of a fraction?

Yes, in some cases, the derivative of a fraction can be simplified. For example, if the numerator and denominator have a common factor, it can be factored out and canceled. Additionally, if the fraction is raised to a power, you can use the power rule to simplify the derivative.

4. What is the significance of the derivative of a fraction?

The derivative of a fraction can tell us important information about the behavior of the original function. For example, the derivative can tell us where the function is increasing or decreasing, where it has maximum or minimum points, and where it is concave up or concave down.

5. Can the derivative of a fraction be negative?

Yes, the derivative of a fraction can be negative. This indicates that the function is decreasing at that point. It is important to note that the sign of the derivative only tells us about the direction of change, not the actual value of the change.

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