Question

$d^{20} / d x^{20}(2 \cos x \cos 3 x)=$
(1) $2^{20}\left(\cos 2 x-2^{20} \cos 4 x\right)$
(2) $2^{20}\left(\cos 2 x+2^{20} \cos 4 x\right)$
(3) $2^{20}\left(\sin 2 x+2^{20} \sin 4 x\right)$
(4) $2^{20}\left(\sin 2 x-2^{20} \sin 4 x\right)$

Answer 1

Hint: Here we have to differentiate the function 20 times with respect to $x$. But it is not a practical way to solve a problem easily. So we are using the formula for the nth derivative and finding the differentiable value up to 20 times.

Formula Used:
The general formula $2 \cos A \cos B=\cos (A+B)+\cos (A-B)$

Complete step by step Solution:
Let $y=(2 \cos x \cos 3 x)=$
We know $2 \cos A \cos B=\cos (A+B)+\cos (A-B)$
$\text { So } y=\cos (x+3 x)+\cos (x-3 x)$
$=\cos 4 x+\cos 2 x(\text { since } \cos (-x)=\cos x)$
The rate of change of $x$ with respect to y is defined by $dy/dx$ if x and y are two variables. The universal representation of a function's derivative is given by the equation $f'(x) = dy/dx$, where $y = f(x)$ is any function.
Differentiate with respect to $x$
$d y / d x=-4 \sin 4 x-2 \sin 2 x$
$=-2(\sin 2 x+2 \sin 4 x)$
Differentiating again
$d^{2} y / d x^{2}=-2(2 \cos 2 x+8 \cos 4 x)$
$=-4\left(\cos 2 x+2^{2} \cos 4 x\right)$
$d^{3} y / d x^{3}=-4(-2 \sin 2 x-16 \sin 4 x)$
$=8 \sin 2 x+64 \sin 4 x$
$=2^{3} \sin 2 x+4^{3} \sin 4 x$
$=4^{3} \sin 4 x+2^{3} \sin 2 x$
$d^{4} y / d x^{4}=4^{4} \cos 4 x+2^{4} \cos 2 x$
We cant differentiate the term upto 20 times
So we are taking the nth derivative of the function
Any one of several higher-order derivatives of a function is referred to as the nth derivative.
The first derivative is obtained by taking the function's derivative once. You can obtain the second derivative by differentiating the new function once again. The third derivative, fourth derivative, or fifth derivative are obtained by using the differentiation rules a third, fourth, or fifth time, accordingly. The formula for each additional derivative of a function is known as the nth derivative.
Similarly
$d^{4 n} y / d x^{4 n}=4^{4 n} \cos 4 x+2^{4 n} \cos 2 x$
Put $n=5$
$\text { So } d^{20} y / d x^{20}=4^{20} \cos 4 x+2^{20} \cos 2 x$
$=2^{20}\left(2^{20} \cos 4 x+\cos 2 x\right)$
$=2^{20}\left(\cos 2 x+2^{20} \cos 4 x\right)$

Hence, the correct option is 2.

Note: Apart from integration, differentiation is one of the two key ideas of calculus. A technique for determining a function's derivative is differentiation. Mathematicians use a procedure called differentiation to determine a function's instantaneous rate of change based on one of its variables. The most typical illustration is velocity, which is the rate at which a distance changes in relation to time. Finding an antiderivative is the opposite of differentiation.