Question

Solve the given question in a detail manner:
Find \[\dfrac{{dy}}{{dx}}\], if \[x - y = \pi \].

Answer 1

Hint: Take the given equation and differentiate the equation on both the sides with respect to \[x\]. Then we use the distributive law to differentiate each variable term. The differentiation of the constant is zero. Like terms are cancelled and we write the needed value on one side and the rest other terms on the other side to obtain the final equation.

Complete step-by-step solution:
Given equation;
\[x - y = \pi \]
To get \[\dfrac{{dy}}{{dx}}\], differentiate the equation both the sides with respect to \[x\]
\[ \Rightarrow \dfrac{d}{{dx}}\left( {x - y} \right) = \dfrac{d}{{dx}}\pi \]
Using the distributive rule, we can obtain;
\[ \Rightarrow \dfrac{{dx}}{{dx}} - \dfrac{{dy}}{{dx}} = \dfrac{{d\pi }}{{dx}}\]
\[\dfrac{{dx}}{{dx}}\] is cancelled. Which means, we get;
\[ \Rightarrow 1 - \dfrac{{dy}}{{dx}} = \dfrac{{d\pi }}{{dx}}\]
\[\dfrac{{d\pi }}{{dx}}\] is differentiated. Since \[\pi \] is a constant, the differentiation is \[0\]
To obtain \[\dfrac{{dy}}{{dx}}\], we take the rest of the other terms to the right-hand side. We get;
\[ \Rightarrow 1 = \dfrac{{dy}}{{dx}}\]
This can also be written as;
\[\dfrac{{dy}}{{dx}} = 1\]

Therefore, we have the value of \[\dfrac{{dy}}{{dx}}\] is 1.

Note: Differentiation is the process of finding a derivative. It is a rate of change of a function. The differentiation of an equation can be done using only three basic functions. The purely algebraic manipulations, basic derivatives, four rules of operation and just a knowledge of how to manipulate functions. There are three basic derivatives. They are;
1. Algebraic function: \[\dfrac{{d{{\left( x \right)}^n}}}{{dx}} = n{x^{n - 1}}\]
2. Trigonometric functions: \[\dfrac{{d\left( {\sin x} \right)}}{{dx}} = \cos x\]
3. Exponential functions: \[\dfrac{{d\left( {{e^x}} \right)}}{{dx}} = {e^x}\]
There are various rules and properties of differentiating a function. Here, in this question, we use the distributive law of differentiation to solve within the parenthesis. The differentiation of a constant is zero because when a constant is derived, there is no variable for the differentiation to be applied.