Question

How do you differentiate $y = \dfrac{6}{x}$ ?

Answer 1

We are given a function that involves the reciprocal of x so the given function is in terms of x. We have to differentiate this function, so we must know what differentiation actually is. Differentiation is defined as the process of dividing a whole quantity into very small ones, in the given question we have to differentiate y with respect to x. Usually, the rate of change of something is observed over a specific duration of time, but differentiation is used when we have to find the instantaneous rate of change of a quantity, it is represented as $\dfrac{{dy}}{{dx}}$ , in the expression $\dfrac{{dy}}{{dx}}$ , $dy$ represents a very small change in the quantity and $dx$ represents the small change in the quantity with respect to which the given quantity is changing.

Complete step by step answer:
We are given that $y = \dfrac{6}{x}$ and we have to differentiate the function y.
We know that the differentiation of the product of a constant and a function is equal to the product of the constant and the derivative of the function. So,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(\dfrac{6}{x}) \\\ \Rightarrow \dfrac{{dy}}{{dx}} = 6\dfrac{d}{{dx}}(\dfrac{1}{x}) \\\$
The derivative of ${x^n}$ is $n{x^{n - 1}}$ so,

$$\Rightarrow \dfrac{{dy}}{{dx}} = 6(\dfrac{{ - 1}}{{{x^2}}}) \\\ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - 6}}{{{x^2}}} \\\$$

Hence the derivative of $y = \dfrac{6}{x}$ is equal to $\dfrac{{ - 6}}{{{x^2}}}$ .

Note: While differentiating an equation, we must rearrange the equation first, so that the one side contains the variable with respect to which we are differentiating and the other side should contain the variable whose derivative we have to find. In the given question, we have a function of x, so by putting different values of x, we can find the instantaneous change in x at that particular value.