Using the first principle, prove that: `d/(dx)(f(x)g(x))=f(x)d/(dx)(g(x))+g(x)d/(dx)(f(x))`

Using the first principle, prove that d/(dx)(1/(f(x)))=(-f^(prime)(x))/([f(x)]^2)

Using first principles, prove that d/(dx){1/(f(x))}=-(f^(prime)(x))/({f(x)}^2)

Using first principles, prove that (d)/( dx) ((1)/( f(x))) = (-f' (x))/( {f (x) }^2) .

If f(x)a n dg(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)=(x^2)/(|x|) , write d/(dx)(f(x))

If f(x)a n dg(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/x{f(x)}-g(x)d/x{g(x)})/([g(x)]^2)

If f(x)a n dg(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)}

Differentiate by first principle f(x)=sqrt(3x+4)

Statement-1: int(sinx)^x(xcotx+logsinx)dx=x(sinx)^x Statement-2: d/dx(f(x))^(g(x))=(f(x))^(g(x))d/dx[g(x)logf(x)] (A) Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1. (C) Statement-1 is True, Statement-2 is False. (D) Statement-1 is False, Statement-2 is True.

f_n(x)=e^(f_(n-1)(x)) for all n in Na n df_0(x)=x ,t h e n d/(dx){f_n(x)} is (a) (f_n(x)d)/(dx){f_(n-1)(x)} (b) f_n(x)f_(n-1)(x) (c) f_n(x)f_(n-1)(x).......f_2(x)dotf_1(x) (d)none of these