Question

Вычислите вторую производную (d^2 z)/(dx^2 ) функции :


z=e^x sin⁡x.
​

Answer 1

$z = e^x\cdot\sin(x)$

$\frac{d z}{dx} = (e^x\cdot\sin(x))' = (e^x)'\cdot\sin(x) + e^x\cdot (\sin(x))' =$

$= e^x\cdot\sin(x) + e^x\cdot\cos(x) = e^x\cdot (\sin(x) + \cos(x))$

$\frac{d^2z}{dx^2} = \frac{d}{dx} \left( e^x\cdot(\sin(x) + \cos(x)) \right) =$

$= (e^x)'\cdot (\sin(x) + \cos(x)) + e^x\cdot (\sin(x) + \cos(x))' =$

$= e^x\cdot (\sin(x) + \cos(x)) + e^x\cdot ( (\sin(x))' + (\cos(x))' ) =$

$= e^x\cdot (\sin(x) + \cos(x)) + e^x\cdot ( \cos(x) - \sin(x) ) =$

$= e^x\cdot( \sin(x) + \cos(x) + \cos(x) - \sin(x) ) =$

$= e^x\cdot( 2\cos(x) ) = 2\cdot e^x \cdot\cos(x)$