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-x^2)=-2x

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

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)))

Newton-Leinbnitz's formula states that d/dx(int_(phi(x))^(psi(x)) f(t)dt)=f(psi(x)){d/dxpsi(x)}-f(phi(x)){d/dxphi(x)} Answer the following questions based on above passage: Let f(x) be a differentiable function and f(1)=2, If lim_(xrarr1)int_2^(f(x)) (2t)/(x-1)dt=4 , then the value of f'(1) is equal to

Newton-Leinbnitz's formula states that d/dx(int_(phi(x))^(psi(x)) f(t)dt)=f(psi(x)){d/dxpsi(x)}-f(phi(x)){d/dxphi(x)} Answer the following questions based on above passage: If lim_(xrarr0)int_0^(x) (t^2dt)/((x-sinx)sqrt(a+t))=1 , then the value of a is

Newton-Leinbnitz's formula states that d/dx(int_(phi(x))^(psi(x)) f(t)dt)=f(psi(x)){d/dxpsi(x)}-f(phi(x)){d/dxphi(x)} Answer the following questions based on above passage: lim_(xrarr0)int_0^(x^2) ((cost)^2dt)/((xsinx))=

If f(x) = logx, then d/(dx)f(logx) =

The value of (d)/(dx)5^(f(x)) is -

The function y=f(x) is the solution of the differential equation (dy)/(dx)+(xy)/(x^(2)-1)=(x^(4)+2x)/(sqrt(1-x^(2))) in (-1, 1) satisfying f(0)=0 . Then underset((-sqrt(3))/(2))overset((sqrt(3))/(2))intf(x)dx is

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