Question

If $y = \left( {1 + x} \right)\left( {1 + {x^2}} \right)\left( {1 + {x^4}} \right)........\left( {1 + {x^{2n}}} \right)$ then the value of ${\left( {\dfrac{{dy}}{{dx}}} \right)_x} = 0$ is
$\eqalign{
  & 1)0 \cr
  & 2) - 1 \cr
  & 3)1 \cr
  & 4)2 \cr} $

Answer 1

Hint: The given question is in the form of a sequence. To solve this question, we will take logarithms on both sides of the equation. Then we can differentiate. Lastly, we need to find the value of the differentiated equation at $0$. We need to differentiate the equation only once.
The formulas used to solve this problem are:
$$\eqalign{
  & \dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}} \cr
  & \dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x} \cr} $$

Complete step-by-step answer:
Let us consider the given equation, $y = \left( {1 + x} \right)\left( {1 + {x^2}} \right)\left( {1 + {x^4}} \right)........\left( {1 + {x^{2n}}} \right)$
Now, let us take $\log $of these functions on both sides.
We get,
$\log y = \log \left( {1 + x} \right) + \log \left( {1 + {x^2}} \right) + \log \left( {1 + {x^4}} \right) + ........ + \log \left( {1 + {x^{2n}}} \right)$
Now, differentiating with respect to $x$,
$ \Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \dfrac{1}{{1 + x}} + \dfrac{{2x}}{{1 + {x^2}}} + \dfrac{{4{x^3}}}{{1 + {x^4}}} + ........ + \dfrac{{2n{x^{2n - 1}}}}{{1 + {x^{2n}}}}$
By simplifying the terms, we get,
$ \Rightarrow \dfrac{{dy}}{{dx}} = y\left( {\dfrac{1}{{1 + x}} + \dfrac{{2x}}{{1 + {x^2}}} + \dfrac{{4{x^3}}}{{1 + {x^4}}} + ........ + \dfrac{{2n{x^{2n - 1}}}}{{1 + {x^{2n}}}}} \right)$
This is the most simplified version of the derivative. So, we stop here.
Now, we substitute $x = 0$ for the above equation
$ \Rightarrow \dfrac{{dy}}{{dx}} = y\left( {\dfrac{1}{{1 + 0}} + \dfrac{{2 \times 0}}{{1 + {0^2}}} + \dfrac{{{{4 \times 0}^3}}}{{1 + {0^4}}} + ........ + \dfrac{{2n.{0^{2n - 1}}}}{{1 + {0^{2n}}}}} \right)$
That is,
$\eqalign{
  & \Rightarrow {\dfrac{{dy}}{{dx}}_{x = 0}} = 1(1 + 0 + 0 + ...... + 0) \cr
  & \Rightarrow {\dfrac{{dy}}{{dx}}_{x = 0}} = 1 \cr} $
Therefore, the final answer is $1$
Hence, the correct option is (3) which gives us the correct answer.
So, the correct answer is “Option 3”.

Note: The differentiation is also called as the process of finding a derivative gives us the rate of change of one function with respect to another. Each of the trigonometric functions, algebraic and the arithmetic functions have their own derivatives. The derivatives are used to find out the maximum values and the minimum values of the particular functions.