if (e^{y}-e^{-y}=x - x^{3}), then the value of (\frac{dy}{dx}) at the point ((0,1)) is\na (-\frac{1}{e}) \nb…

Explanation:

Step1: Differentiate the given function

Given (y = e^{y}-x - x^{3}). Differentiate both sides with respect to (x) using implicit - differentiation. The derivative of (y) with respect to (x) is (\frac{dy}{dx}), the derivative of (e^{y}) with respect to (x) is (e^{y}\frac{dy}{dx}) (by the chain - rule), the derivative of (x) with respect to (x) is (1), and the derivative of (x^{3}) with respect to (x) is (3x^{2}). So we have (\frac{dy}{dx}=e^{y}\frac{dy}{dx}-1 - 3x^{2}).

Step2: Isolate (\frac{dy}{dx}) terms

Rearrange the equation (\frac{dy}{dx}-e^{y}\frac{dy}{dx}=-1 - 3x^{2}). Factor out (\frac{dy}{dx}) on the left - hand side: (\frac{dy}{dx}(1 - e^{y})=-1 - 3x^{2}). Then (\frac{dy}{dx}=\frac{-1 - 3x^{2}}{1 - e^{y}}).

Step3: Substitute the point ((0,1))

Substitute (x = 0) and (y = 1) into (\frac{dy}{dx}). We get (\frac{dy}{dx}\big|_{(0,1)}=\frac{-1-3\times0^{2}}{1 - e^{1}}=\frac{-1}{1 - e}=\frac{1}{e - 1}).

Answer:

B. (\frac{1}{e - 1})