Find the derivative of cosx with respect to x using the first principle.
Compute the derivative:
As we know, the formula for the first principle,
f'(x)=limh→0[f(x+h)-f(x)h]
We can write that
f'(x)=limh→0cosx+h-cosxh
f'(x)=limh→0-2hsinx+h+x2sinx+h-x2∵cosA-cosB=-2sinA+B2sinA-B2
f'(x)=limh→0sinx+h+x2sinx+h-x2x+h-x2x+h-x2-2h
f'(x)=limsinh→0x+h+x2limh→0sinx+h-x2x+h-x2limh→0x+h12-x12h-1
f'(x)=-sinx1×limh→012x+h-01UseLHospitalruledifferentiatew.r.th
f'(x)=-sinx2x
Hence the required derivative is -sinx2x