Question:

If ey(x+1)=1e^y(x+1)=1,show that d2ydx2=(dydx)2\frac{d^2y}{dx^2}=(\frac{dy}{dx})^2

Updated On: Feb 19, 2024
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Solution and Explanation

The given relationship is,ey(x+1)=1e^y(x+1)=1
ey(x+1)=1e^y(x+1)=1
ey=1x+1⇒e^y=\frac{1}{x+1}
Taking logarithm on both the sides, we obtain
y=log1(x+1)y=log\frac{1}{(x+1)}
Differentiating this relationship with respect to xx, we obtain
dydx=(x+1)ddx(1x+1)=(x+1).1(x+1)2=1(x+1)\frac{dy}{dx}=(x+1)\frac{d}{dx}(\frac{1}{x+1})=(x+1).\frac{-1}{(x+1)^2}=\frac{-1}{(x+1)}
d2ydx2=ddx(1(x+1))=(1(x+1)2)∴\frac{d^2y}{dx^2}=\frac{-d}{dx}(\frac{1}{(x+1)})=-(\frac{-1}{(x+1)^2})=1(x+1)2=\frac{1}{(x+1)^2}
d2ydx2=(1(x+1))2⇒\frac{d^2y}{dx^2}=(\frac{-1}{(x+1)})^2
d2ydx2=(dydx)2\frac{d^2y}{dx^2}=(\frac{dy}{dx})^2
Hence, proved.
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Concepts Used:

Second-Order Derivative

The Second-Order Derivative is the derivative of the first-order derivative of the stated (given) function. For instance, acceleration is the second-order derivative of the distance covered with regard to time and tells us the rate of change of velocity. 

As well as the first-order derivative tells us about the slope of the tangent line to the graph of the given function, the second-order derivative explains the shape of the graph and its concavity. 

The second-order derivative is shown using f’’(x) or d2ydx2f’’(x)\text{ or }\frac{d^2y}{dx^2}.