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)

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

Suppose f(x)=(d)/(dx)(e^(x)+2) . Find intf(x)dx

If f(x)=x^(3)+3x+4 and g is the inverse function of f(x), then the value of (d)/(dx)((g(x))/(g(g(x)))) at x = 4 equals

Suppose f(x)=(d)/(dx)(e^(x)+2) . Find int(f(x)+x^(2))dx

If (d)/(dx)f(x)=4x^(3)-(3)/(x^(4)) , then f(x) is

If a curve is represented parametrically by the equation x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(2)))

Statement 1: If differentiable function f(x) satisfies the relation f(x)+f(x-2)=0AAx in R , and if ((d/(dx)f(x)))_(x=a)=b ,t h e n((d/(dx)f(x)))_(a+4000)=bdot Statement 2: f(x) is a periodic function with period 4.

Let the function ln f(x) is defined where f(x) exists for x geq 2 and k is fixed positive real numbers prove that if d/(dx) (x.f(x)) geq -k f(x) then f(x) geq Ax^(-1-k) where A is independent of x.

If intsinx(d)/(dx)("sec x")dx=f(x)-g(x)+C , then