Question

If $y = \log \left( {\cos ecx - \cot x} \right)$, then find the value of $\dfrac{{dy}}{{dx}}$.
A) $\cos ecx + \cot x$
B) $\cot x$
C) $\sec x + \tan x$
D) $\cos ecx$

Answer 1

In this question, we are given an equation and we have been asked to differentiate the given equation. Differentiate both the sides with respect to x. then, use chain rule on RHS to differentiate further. Take out one term common out of it such that there are like terms in the numerator and denominator and then cancel the like terms. You will get your answer.

Complete step-by-step solution:
We are given an equation $y = \log \left( {\cos ecx - \cot x} \right)$. We have to find the $\dfrac{{dy}}{{dx}}$ of the equation.
$\Rightarrow$ $y = \log \left( {\cos ecx - \cot x} \right)$
Differentiating both the sides with respect to x,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{\left( {\cos ecx - \cot x} \right)}}\dfrac{{d\left( {\cos ecx - \cot x} \right)}}{{dx}}$ …. $\left( {\dfrac{{d\left( {\log x} \right)}}{{dx}} = \dfrac{1}{x}} \right)$
Using chain rule to find the differentiation of 2nd part,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\left( { - \cos ecx\cot x + \cos e{c^2}x} \right)}}{{\left( {\cos ecx - \cot x} \right)}}$ …. $\left( {\dfrac{{d\left( {\cos ecx} \right)}}{{dx}} = - \cos ecx\cot x,\dfrac{{d\left( {\cot x} \right)}}{{dx}} = - \cos e{c^2}x} \right)$
Now, we will take common cosec x from the numerator,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\cos ec\left( { - \cot x + \cos ecx} \right)}}{{\left( {\cos ecx - \cot x} \right)}}$
Rearranging the terms in the bracket,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\cos ec\left( {\cos ecx - \cot x} \right)}}{{\left( {\cos ecx - \cot x} \right)}}$
Since, there are like terms in the numerator and denominator, we will cancel them and it will give us our answer.
$\Rightarrow \dfrac{{dy}}{{dx}} = \cos ecx$

Therefore, our required answer is option D) $\cos ecx$.

Note: hain rule is a very important technique in differentiation. It helps us in differentiating the composite functions. In such functions, we identify the inner and outer functions first. It is very important to identify them correctly. For example: ${\left( {7x - 9{x^2}} \right)^5}$ is a composite function. In this, $7x - 9{x^2}$ is an inner function and we have to assume this inner function as x. Now, our outer function has become ${x^5}$.
So, first we differentiate the outer function. Then, we differentiate the inner function.
A common mistake students make is that they only differentiate the outer function and forget to differentiate the inner function. So, always do the question step by step.