Differentiate -5/3x: "Why is Derivative 5/3x^2?

In summary: The power rule "works" for negative powers because we have proven it by using the quotient rule.In summary, the derivative of f(x)=-5/3x is simply 5/3x^2 because of the power rule for differentiating functions like x^n. To differentiate the function, we rewrite it as -5/3 * 1/x and then use the power rule to get 5/3 * 1/x^2, which can also be written as -5/3x^2. Alternatively, we can use the quotient rule to prove the power rule for negative powers, which is necessary before using the power rule for negative powers.
  • #1
stat643
16
0
i can differentiate most other simple functions.. .though can someone please help me to understand why the derivative of f(x)=-5/3x is simply 5/3x^2?
 
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  • #2
Do you know the power rule for differentiating functions like [itex]x^n[/itex]? (Hint: [itex]1/x = x^{?}[/itex].)
 
  • #3
Its basically following the rule:

[tex] \frac{d}{dx} ax^n = anx^{n-1} [/tex]

where in your example [itex]a=\frac{-5}{3}[/itex] and [itex] n = -1[/itex].

EDIT: puppy interupted hence late reply :tongue:
 
Last edited:
  • #4
1/x = x^-1 and yes i can differentiate functions in the form x^n
 
  • #5
stat643 said:
1/x = x^-1 and yes i can differentiate functions in the form x^n
Then you should be able to solve this as that's all that's needed.
 
  • #6
oh i see , i just didnt realize n was -1, but obviously it is wen its on the denominator, problem solved. cheers
 
  • #7
F(x) = [tex]\frac{-5}{3x}[/tex]

The first thing you got to do is re-write the function so we get:

F(x) = [tex]\frac{-5}{3}[/tex] . [tex]\frac{1}{x}[/tex]

F(x) = [tex]\frac{-5}{3}[/tex] . x[tex]^{-1}[/tex]

after re-writing the function, find the most appropriated way to derivate the function in this case it would be: [tex]\frac{d}{dx}[/tex] [u[tex]^{n}[/tex]] = nu[tex]^{n-1}[/tex] u'

Using this rule we get:

F(x) = [tex]\frac{-5}{3}[/tex] . x[tex]^{-1}[/tex]

F'(x) = (-1) . [tex]\frac{-5}{3}[/tex] . x[tex]^{-2}[/tex] (1) , then simplify

F'(x) = [tex]\frac{5}{3}[/tex] . x[tex]^{-2}[/tex], then re-write

F'(x) = [tex]\frac{5}{3}[/tex] . [tex]\frac{1}{x^{2}}[/tex]

F'(x) = [tex]\frac{-5}{3x^{2}}[/tex]
 
  • #8
You don't have to use the power rule for negative powers, you can use the quotient rule: -5/3x= f(x)/g(x) with f(x)= -5 and g(x)= 3x.
(f/g)'= (f'g- fg')(g2)

Since f'= 0 and g'= 3, that gives (-5/3x)'= ((0)(3x)- (-5)(3))/(9x2= 5/3x2.

The reason I mention that is that before you can use the power rule for negative powers you have to prove it for negative powers- and you do that by using the quotient rule:
x-n= 1/xn= f/g with f(x)= 1, g(x)= xn. f'= 0, g'= nxn-1 so (x-n)'= (1/xn)'= ((0)(xn)- 1(nxn-1)/x2n= n xn-1/x2n= n x(n-1)- 2n= n x-n-1.
 

1. Why is the derivative of -5/3x equal to 5/3x^2?

The derivative of a function is a measure of its rate of change at a specific point. In this case, the function -5/3x represents a straight line with a slope of -5/3. When finding the derivative, we use the power rule which states that for a function of the form y = ax^n, the derivative is given by dy/dx = anx^(n-1). Applying this rule to -5/3x, we get a derivative of -5/3x^(1-1) = -5/3x^0 = -5/3. This is the constant slope of the original function. However, since the derivative is a function of x, we can rewrite it as -5/3x^1 = -5/3x, which is equivalent to 5/3x^2.

2. Can the derivative of -5/3x be simplified further?

No, the derivative of -5/3x is already in its simplest form. It is important to note that the derivative is only a measure of the slope of the function at a specific point and may not represent the entire behavior of the function.

3. How do you find the derivative of -5/3x?

To find the derivative of -5/3x, we use the power rule which states that for a function of the form y = ax^n, the derivative is given by dy/dx = anx^(n-1). In this case, we have a constant value of -5/3, so the derivative is simply -5/3x^(1-1) = -5/3x^0 = -5/3.

4. Is the derivative of -5/3x the same as the derivative of 5/3x?

No, the derivatives of -5/3x and 5/3x are not the same. The negative sign in front of -5/3x represents a negative slope, while the positive sign in front of 5/3x represents a positive slope. This is reflected in their respective derivatives, which are -5/3 and 5/3, respectively.

5. Can the derivative of -5/3x be negative?

Yes, the derivative of -5/3x can be negative. As mentioned before, the negative sign in front of -5/3x represents a negative slope. This means that as x increases, the value of -5/3x decreases, resulting in a negative derivative. However, the derivative can also be positive or zero, depending on the value of x and the function's behavior.

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