How do you differentiate #f(x)=(cosx+sinx)(lnx-x)# using the product rule?

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Answer 1 · 2017-12-07T02:51:41

#d/dxf(x)=(cosx+sinx)(1/x-1)+(lnx-x)(-sinx+cosx)#

#f(x)=(cosx+sinx)(lnx-x)#

Here #f(x)# is the product of two functions.

Therefore to differentiate #f(x)# we will use the product rule

The product rule says that #->#

#d/dxU*V=Ud/dxV+Vd/dxU#

Now back to the question, we will differentiate both sides with respect to #x#

#d/dxf(x)=d/dx[(cosx+sinx)(lnx-x)]#

#d/dxf(x)=(cosx+sinx)d/dx(lnx-x)+(lnx-x)d/dx(cosx+sinx)#

So therefore,

#d/dxf(x)=(cosx+sinx)(1/x-1)+(lnx-x)(-sinx+cosx)#