Question:
Find the second order derivatives of the function
\(e^{6x}cos3x\)
Find the second order derivatives of the function
\(e^{6x}cos3x\)
\(e^{6x}cos3x\)
Updated On: Sep 12, 2023
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Solution and Explanation
The correct answer is \(=9e^{6x}(3cos3x-4sin3x)\)
Let \(y=e^{6x}cos3x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(e^{6x}cos3x)=cos3x.\frac{d}{dx}(e^{6x})+e^{6x}.\frac{d}{dx}(cos3x)\)
\(=cos3x.e^{6x}.\frac{d}{dx}(6x)+e^{6x}.(-sin3x).\frac{d}{dx}(3x)\)
\(=6e^{6x}cos3x-3e^{6x}sin3x ......(1)\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[6e^{6x}cos3x-3e^{6x}sin3x]=6.\frac{d}{dx}(e^{6x}cos3x)-3.\frac{d}{dx}(e^{6x}sin3x)\)
\(=6.[6e^{6x}cos3x-3e^{6x}sin3x]-3.[sin3x.\frac{d}{dx}(e^{6x})+e^{6x}.\frac{d}{dx}(sin3x)]\) [using(1)]
\(=36e^{6x}cos3x-18e^{6x}sin3x-3[sin3x.e^{6x}.6+e^{6x}.cos3x.3]\)
\(=36e^{6x}cos3x-18e^{6x}sin3x-18e^{6x}sin3x-9e^{6x}cos3x\)
\(=27e^{6x}cos3x-36e^{6x}sin3x\)
\(=9e^{6x}(3cos3x-4sin3x)\)
Let \(y=e^{6x}cos3x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(e^{6x}cos3x)=cos3x.\frac{d}{dx}(e^{6x})+e^{6x}.\frac{d}{dx}(cos3x)\)
\(=cos3x.e^{6x}.\frac{d}{dx}(6x)+e^{6x}.(-sin3x).\frac{d}{dx}(3x)\)
\(=6e^{6x}cos3x-3e^{6x}sin3x ......(1)\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[6e^{6x}cos3x-3e^{6x}sin3x]=6.\frac{d}{dx}(e^{6x}cos3x)-3.\frac{d}{dx}(e^{6x}sin3x)\)
\(=6.[6e^{6x}cos3x-3e^{6x}sin3x]-3.[sin3x.\frac{d}{dx}(e^{6x})+e^{6x}.\frac{d}{dx}(sin3x)]\) [using(1)]
\(=36e^{6x}cos3x-18e^{6x}sin3x-3[sin3x.e^{6x}.6+e^{6x}.cos3x.3]\)
\(=36e^{6x}cos3x-18e^{6x}sin3x-18e^{6x}sin3x-9e^{6x}cos3x\)
\(=27e^{6x}cos3x-36e^{6x}sin3x\)
\(=9e^{6x}(3cos3x-4sin3x)\)
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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 \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).