Question

Find the second order derivatives of the function.e^x.sin5x

Answer 1

Let $\bf{y=e^x.sin5x}$
now differentiate y with respect to x,
$\bf{\frac{dy}{dx}=\frac{d(e^x.sin5x)}{dx}}\\\\=\bf{e^x.\frac{d(sin5x)}{dx}+sin5x\frac{d(e^x)}{dx}}\\\\=\bf{e^x.5cos5x+sin5x.e^x}$

so, $\bf{\frac{dy}{dx}=e^x(5cos5x+sin5x)}$
differentiate $\bf{\frac{dy}{dx}}$ once again,

$\bf{\frac{d^2y}{dx^2}=\frac{d\{e^x(5cos5x+sin5x)\}}{dx}}\\\\=\bf{e^x\frac{d(5cos5x+sin5x)}{dx}+(5cos5x+sin5x)\frac{d(e^x)}{dx}}\\\\=\bf{e^x(25\times-sin5x+5cos5x)+(5cos5x+sin5x)e^x}\\\\=\bf{10e^x.cos5x-24e^x.sin5x}$

hence, d²y/dx² = 10e^x.cos5x - 24e^x.sin5x
= e^x(10cos5x - 24sin5x)