Question

Find the second order derivatives of the function
\(e^xsin5x\)

Answer 1

The correct answer is \(2e^x(5cos5x-12sin5x)\)
Let \(y=e^xsin5x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(e^xsin5x)=sin5x.\frac{d}{dx}(e^x)+e^x\frac{d}{dx}(sin5x)\)
\(=sin5x.e^x+e^x.cos5x\frac{d}{dx}(5x)=e^xsin5x+e^xcos5x.5\)
\(=e^x(sin5x+5cos5x)\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[e^x(sin5x+5cos5x)]\)
\(=(sin5x+5cos5x).\frac{d}{dx}(e^x)+e^x.\frac{d}{dx}(sin5x+5cos5x)\)
\(=(sin5x+5cos5x)e^x+e^x[cos5x.\frac{d}{dx}(5x)+5(-sin5x).\frac{d}{dx}(5x)]\)
\(=e^x(sin5x+5cos5x)+e^x(5cos5x-25sin5x)\)
Then, \(e^x(10cos5x-24sin5x)=2e^x(5cos5x-12sin5x)\)