Q. 325.0( 2 Votes )

If y = ex cosx, prove that   (CBSE 2012)

Answer :

Basic idea:


√Second order derivative is nothing but derivative of derivative i.e.


√The idea of chain rule of differentiation: If f is any real-valued function which is the composition of two functions u and v, i.e. f = v(u(x)). For the sake of simplicity just assume t = u(x)


Then f = v(t). By chain rule, we can write the derivative of f w.r.t to x as:



√Product rule of differentiation-


√Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..


Let’s solve now:


Given,


y=ex cos x


TO prove :



Clearly from the expression to be proved we can easily observe that we need to just find the second derivative of given function.


Given, y = ex cos x


We have to find


As


So lets first find dy/dx and differentiate it again.



Let u = ex and v = cos x


As, y = u*v


Using product rule of differentiation:




[ ]


Again differentiating w.r.t x:




Again using the product rule :




[


[ –sin x = cos (x + π/2)]


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