Question

How do you find the derivative of\[f(x)=-\dfrac{9}{x}\]?

Answer 1

Hint: In this question, we will find the derivation of a function \[-\dfrac{9}{x}\]. To solve this question, we have to remember a formula in differentiation. The formula is the quotient rule of the differentiation.
The quotient rule says that if there are two functions such as u(x) and v(x).
The derivation of \[\dfrac{u(x)}{v(x)}\] is
\[\dfrac{d}{dx}\left( \dfrac{u(x)}{v(x)} \right)=\dfrac{v(x)\dfrac{d\left( u(x) \right)}{dx}-u(x)\dfrac{d\left( v(x) \right)}{dx}}{{{\left( v(x) \right)}^{2}}}\]

Complete step by step answer:
Let us solve this question.
We have to find the derivation of \[-\dfrac{9}{x}\].
For that we will have to apply a formula of derivation. That is the quotient rule of differentiation.
The division rule is:
\[\dfrac{d}{dx}\left( \dfrac{u(x)}{v(x)} \right)=\dfrac{v(x)\dfrac{d\left( u(x) \right)}{dx}-u(x)\dfrac{d\left( v(x) \right)}{dx}}{{{\left( v(x) \right)}^{2}}}\]
In the equation \[f(x)=-\dfrac{9}{x}\] , we have u(x)=-9 and v(x)=x
Then the derivation of f(x) is
\[\begin{align}
  & \dfrac{d\left( f(x) \right)}{dx}=\dfrac{d\left( \dfrac{-9}{x} \right)}{dx}=\dfrac{x\times \dfrac{d(-9)}{dx}-(-9)\times \dfrac{d(x)}{dx}}{{{\left( x \right)}^{2}}} \\
 & \\
\end{align}\]
As we know that differentiation of any constant is 0 and differentiation of x with respect to x is 1. Then, we can write the above equation as
\[\Rightarrow \dfrac{d\left( f(x) \right)}{dx}=\dfrac{x\times 0+9\times 1}{{{x}^{2}}}\]
\[\Rightarrow \dfrac{d\left( f(x) \right)}{dx}=\dfrac{9}{{{x}^{2}}}\]

Note: For solving this question, we should know how to find the derivative of any function. And also remember the differentiation formula like, additive rule, subtractive rule, multiplicative rule, and quotient rule of differentiation.
The formulas are:
Suppose, we have two functions u(x) and v(x).
 Then,
Additive rule is : \[\dfrac{d}{dx}\left( u(x)+v(x) \right)=\dfrac{d\left( u(x) \right)}{dx}+\dfrac{d\left( v(x) \right)}{dx}\]
Subtractive rule is : \[\dfrac{d}{dx}\left( u(x)-v(x) \right)=\dfrac{d\left( u(x) \right)}{dx}-\dfrac{d\left( v(x) \right)}{dx}\]
Product rule is : \[\dfrac{d}{dx}\left( u(x)\times v(x) \right)=v(x)\dfrac{d\left( u(x) \right)}{dx}+u(x)\dfrac{d\left( v(x) \right)}{dx}\]
Quotient rule is : \[\dfrac{d}{dx}\left( \dfrac{u(x)}{v(x)} \right)=\dfrac{v(x)\dfrac{d\left( u(x) \right)}{dx}-u(x)\dfrac{d\left( v(x) \right)}{dx}}{{{\left( v(x) \right)}^{2}}}\]
In this question, we have used the quotient rule of derivation in the above solution.
We have an alternate method to solve this question.
Method-2
Whenever, we have to find the derivative of an equation in which the equation has a constant in numerator and the variable x is given in the denominator. Then, the derivation will be in fraction, where negative of constant will be in numerator and the square of x will be in the denominator.
\[\dfrac{d\left( \dfrac{-9}{x} \right)}{dx}=\dfrac{-(-9)}{{{x}^{2}}}=\dfrac{9}{{{x}^{2}}}\]
We can use this method in this type of differentiation.