If y=(1+x)(1+x^2)(1+x^4) dots (1+x^{2^n}), th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $y = (1+x)(1+x^2)(1+x^4) \dots (1+x^{2^n})$.Taking the natural logarithm on both sides:$\ln y = \ln(1+x) + \ln(1+x^2) + \ln(1+x^4) + \dots +

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(C) Given $y = (1+x)(1+x^2)(1+x^4) \dots (1+x^{2^n})$.Taking the natural logarithm on both sides:$\ln y = \ln(1+x) + \ln(1+x^2) + \ln(1+x^4) + \dots + \ln(1+x^{2^n})$.Differentiating both sides with respect to $x$:$\frac{1}{y} \frac{dy}{dx} = \frac{1}{1+x} + \frac{2x}{1+x^2} + \frac{4x^3}{1+x^4} + \dots + \frac{2^n x^{2^n-1}}{1+x^{2^n}}$.Thus, $\frac{dy}{dx} = y \left[ \frac{1}{1+x} + \frac{2x}{1+x^2} + \dots + \frac{2^n x^{2^n-1}}{1+x^{2^n}} \right]$.At $x=0$, $y = (1+0)(1+0) \dots (1+0) = 1$.Substituting $x=0$ into the derivative expression:$\left(\frac{dy}{dx}\right)_{x=0} = 1 \cdot \left[ \frac{1}{1+0} + 0 + 0 + \dots + 0 \right] = 1 \cdot 1 = 1$.

If $y=(1+x)(1+x^2)(1+x^4) \dots (1+x^{2^n})$, then the value of $\left(\frac{dy}{dx}\right)_{x=0}$ is

Similar Questions

An aqueous solution of $6.3 \ g$ of oxalic acid dihydrate is made up to $250 \ mL$. The volume of $0.1 \ N \ NaOH$ required to completely neutralize $10 \ mL$ of this solution is ............. $mL$.

$\lim _{x \rightarrow 0} \frac{\sin x \sin ^{-1} x}{x^2}$ is equal to

$A$ vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles subtended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and $\beta$,then $(\tan \alpha, \tan \beta)$ is equal to

Which of the following molecules has the highest dipole moment?

$A$ $200\, W$ sodium street lamp emits yellow light of wavelength $0.6\, \mu m$. Assuming it to be $50\%$ efficient in converting electrical energy to light,the number of photons of yellow light it emits per second is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module