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

• stat643
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.

#### stat643

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?

Do you know the power rule for differentiating functions like $x^n$? (Hint: $1/x = x^{?}$.)

Its basically following the rule:

$$\frac{d}{dx} ax^n = anx^{n-1}$$

where in your example $a=\frac{-5}{3}$ and $n = -1$.

EDIT: puppy interupted hence late reply :tongue:

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1/x = x^-1 and yes i can differentiate functions in the form x^n

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.

oh i see , i just didnt realize n was -1, but obviously it is wen its on the denominator, problem solved. cheers

F(x) = $$\frac{-5}{3x}$$

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

F(x) = $$\frac{-5}{3}$$ . $$\frac{1}{x}$$

F(x) = $$\frac{-5}{3}$$ . x$$^{-1}$$

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

Using this rule we get:

F(x) = $$\frac{-5}{3}$$ . x$$^{-1}$$

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

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

F'(x) = $$\frac{5}{3}$$ . $$\frac{1}{x^{2}}$$

F'(x) = $$\frac{-5}{3x^{2}}$$

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.