If f(x) and g(x) are differentiable function ; then show that `f(x)pmg(x)` are also differentiable such that `d(f(x)pmg(x))/dx=d(f(x))/dx pm d(g(x))/dx`

If f(x) and g(x) a re differentiate functions, then show that f(x)+-g(x) are also differentiable such that (d)/(dx){f(x)+-g(x)}=(d)/(dx){f(x)}+-(d)/(dx){g(x)}

If f(x) and g(x) are two differentiable functions, show that f(x)g(x) is also differentiable such that (d)/(dx)[f(x)g(x)]=f(x)(d)/(dx){g(x)}+g(x)(d)/(dx){f(x)}

If f(x) and g(f) are two differentiable functions and g(x)!=0, then show trht (f(x))/(g(x)) is also differentiable (d)/(dx){(f(x))/(g(x))}=(g(x)(d)/(pi){f(x)}-g(x)(d)/(x){g(x)})/([g(x)]^(2))

Theorem : (i)Differentiation of a constant function is 0 (ii) Let f(x) be the differentiable function and let c be a constant.Then cf(x) is also differentiable such that d(cf(x))/(dx)=cd((f(x))/(dx)

Let f(x) be a differentiable and let c a be a constant.Then cf(x) is also differentiable such that (d)/(dx){cf(x)}=c(d)/(dx)(f(x))

If y =f(u) is differentiable function of u, and u=g(x) is a differentiable function of x, then prove that y= f [g(x)] is a differentiable function of x and (dy)/(dx)=(dy)/(du)xx(du)/(dx) .

If f(x)dx=g(x) and f^(-1)(x) is differentiable, then intf^(-1)(x)dx equal to

int{f(x)+-g(x)}dx=int f(x)dx+-int g(x)dx

If f is a continuous function on [a;b] and u(x) and v(x) are differentiable functions of x whose values lie in [a;b] then (d)/(dx){int_(u(x))^(v)(x)f(t)dt}=f(v(x))dv(x)/(dx)-f(u(x))du(x)/(dx)

If f and g are continuous functions on [0,a] such that f(x)=f(a-x) and g(x)+g(a-x)=2 then show that int_(0)^(a)f(x)g(x)dx=int_(0)^(a)f(x)dx