​ Please solve
​ Please solve Which of the following function is differentiable at x = 0? (a) cos(lxl) + IXI (b) cos(lxl) — IXI (c) sin(lxl) + IXI (d) sin(lxl) — lx

Dear Student,
Please find below the solution to the asked query:

fx=sinx-x is differentiable at x=0. Let us see howfx=sinx-xf0=0Any function fx is derivable at x=a iflimxa fx-fax-a existsRight hand derivative is given byR.H.D.=limx0+ sinx-x-f0x-0limx0+ sinx-xxFor x>0, mod will open with negatibe sign=limx0+ sinx-xxApply L-H rule=limx0+ cosx-11=cos0-11=1-1=0Left hand derivative is given byL.H.D.=limx0+ sinx-x-f0x-0limx0- sinx-xxFor x<0, mod will open with negatibe sign=limx0- -sinx+xxApply L-H rule=limx0- -cosx+11=-cos0+11=-1+1=0As L.H.D.=R.H.D. hence function is derivable at x=0  

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