Here.
#f(x)=x^2sinx =>f(t)=t^2sint#
We know that,
#f'(x)=lim_(t tox)(f(t)-f(x))/(t-x)#
#=lim_(t tox)(t^2sint-x^2sinx)/(t-x)#
#=lim_(t to x)(t^2sintcolor(red)(-x^2sint+x^2sint)-x^2sinx)/(t-x)...to(1)#
#=lim_(t tox)(sint(t^2-x^2)+x^2(sint-sinx))/(t-x)#
#=lim_(t tox){(sint(t^2-x^2))/(t-x)+(x^2(sint-sinx))/(t-x)}#
=#lim_(t tox){(sint(t-x)(t+x))/((t-x))+(x^2 *2cos((t+x)/2)sin((t-
x)/2))/(t-x)}#
#=sinx(x+x)+2x^2cos((x+x)/2)lim_((t-x)/2 to0){sin((t-x)/2)/(2((t-
x)/2))}#
#=2xsinx+2x^2cosx(1/2),...to[because lim_(theta to0) sintheta/theta=1#
#=2xsinx+x^2cosx#
#=x^2cosx+2xsinx#
#=x(xcosx+2sinx)#
...........................................................................................
Note:
We can take #tocolor(red)(-t^2sinx+t^2sinx# into #(1)#
in place of #tocolor(red)(-x^2sint+x^2sint)#
