Question

If \(e^y = {{e^4} \over {x^4}}\)and \(\frac{{d^2y}}{{dx^2}} = \frac{{A}}{{x^B}}\), then respective values of A and B are _____.
1. 2 and 3
2. 4 and  -2
3. 2 and  -3
4. 4 and 2
5. -4 and 2

Answer 1

Correct Answer - Option 4 : 4 and 2

Concept:

\(\rm \frac{d }{dx} e^{x} = e^{x}\)

Power rule - \(\rm \frac{d }{dx} x^{n} = nx^{n - 1}\)

Derivative of a constant, a:  \(\rm \frac{d }{dx} a = 0\)

Derivative of a constant multiplied with function f: \(\rm \frac{d }{dx} (af) =a \frac{df}{dx}\)

Quotient rule 

\(\rm \frac{d}{dx} \left ( \frac{f}{g} \right ) = \frac{g \frac{d}{dx} (f) - f \frac{d}{dx} (g) }{g^{2}}\)

 

Calculation:

Given

\(\rm e^y = {{e^4} \over {x^4}}\)      ....(i)

Differentiation on both sides w.r.t x

\(\rm e^{y} \frac{dy}{dx} = \frac{-4e^{4}}{x^{5}}\)

put \(\rm e^y = {{e^4} \over {x^4}}\) we get

\(\rm \frac{e^{4}}{x^{4}} \frac{dy}{dx} = \frac{-4e^{4}}{x^{5}}\)

\(\rm \frac{dy}{dx} = \frac{{-4}}{x}\)

Again differentiation on both sides w.r.t x

\(\rm \frac{d^{2}y}{dx^{2}} = \frac{{4}}{x^{2}}\)      ....(ii)

By comparing eq(ii) with \(\rm \frac{{d^2y}}{{dx^2}} = \frac{{A}}{{x^B}}\)

we get the values of A = 4 and B = 2