Question

How do you differentiate \[f\left( x \right) = \dfrac{{\sin x}}{x}\]?

Answer 1

Hint: Here we need to find the derivative of the given function. The given function is in fraction i.e. the numerator of the fraction contains the function which is different from the function in the denominator. So we will use the quotient rule to differentiate the given function. After using the quotient rule, we will use different mathematical operations like addition, subtraction, and division to simplify it further to get the required value.

Formula used:
According to the quotient rule in derivative, if \[f\left( x \right) = \dfrac{{g\left( x \right)}}{{h\left( x \right)}}\] then \[\dfrac{{df}}{{dx}} = \dfrac{{\dfrac{{dg}}{{dx}} \times h\left( x \right) - \dfrac{{dh}}{{dx}} \times g\left( x \right)}}{{{{\left( {h\left( x \right)} \right)}^2}}}\].

Complete step by step solution:
Here we need to differentiate the given function and the given function is \[f\left( x \right) = \dfrac{{\sin x}}{x}\].
We can see that the given function is in the form of \[f\left( x \right) = \dfrac{{g\left( x \right)}}{{h\left( x \right)}}\] i.e. the numerator of the fraction contains the function which is different from the function in the denominator.
So, we will use the quotient rule to differentiate the given function.
We can say that in the given function,
\[\begin{array}{l}g\left( x \right) = \sin x\\h\left( x \right) = x\end{array}\]
On using the quotient rule \[\dfrac{{df}}{{dx}} = \dfrac{{\dfrac{{dg}}{{dx}} \times h\left( x \right) - \dfrac{{dh}}{{dx}} \times g\left( x \right)}}{{{{\left( {h\left( x \right)} \right)}^2}}}\], we get
\[\dfrac{{df\left( x \right)}}{{dx}} = \dfrac{{\dfrac{{d\left( {\sin x} \right)}}{{dx}} \times x - \dfrac{{d\left( x \right)}}{{dx}} \times \sin x}}{{{{\left( x \right)}^2}}}\]
On differentiating the terms, we get
\[ \Rightarrow \dfrac{{df\left( x \right)}}{{dx}} = \dfrac{{\cos x \times x - 1 \times \sin x}}{{{x^2}}}\]
On multiplying the terms, we get
\[ \Rightarrow \dfrac{{df\left( x \right)}}{{dx}} = \dfrac{{x\cos x - \sin x}}{{{x^2}}}\]
We can see this cannot be simplified further as there are no like terms left.

Hence, the required derivative of the given fraction is equal to \[\dfrac{{x\cos x - \sin x}}{{{x^2}}}\].

Note:
Here we have obtained the required derivative of the given fraction. Here the derivative of a given function is defined as the change of the given function with respect to the variable used in the given function. Differentiation is defined as the action of calculating the derivative of a function. Integration is inverse of differentiation and hence, it is called antiderivative. Integration is defined as the summation of all the discrete data.