If \(y=500e^{7x}+600e^{-7x}\),show that \(\frac{d^2y}{dx^2}=49y\)
Solution and Explanation
Then,
\(\frac{dy}{dx}=500.\frac{d}{dx}(e^{7x})+600.\frac{d}{dx}(e^{-7x})\)
\(=500.e^{7x}.\frac{d}{dx}(7x)+600.e^{-7x}.\frac{d}{dx}(-7x)\)
\(=3500e^{7x}-4200e^{-7x}\)
\(\frac{d^2y}{dx^2}=\frac{d}{dx}(3500e^{7x}-4200e^{-7x})\)
\(=3500.\frac{d}{dx}(e^{7x})-4200\frac{d}{dx}(e^{-7x})\)
\(=3500.e^{7x}.\frac{d}{dx}(7x)-4200.e^{-7x}.\frac{d}{dx}(-7x)\)
\(=7\times 3500\times e^{7x}+7\times 4200\times e^{-7x}\)
\(=49\times500e^{7x}+49\times600e^{-7x}\)
\(=49(500e^{7x}+600e^{-7x})\)
\(=49y\)
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 \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).