If `y=u//v," where "u,v," are differentiable functions of x and "vne0," then: "u(dv)/(dx)+v^(2)(dy)/(dx)=`

If u, v, w are three differentiable functions of x , prove that d/dx (uvw) = ((du)/dx)vw + u ((dv)/dx) w + uv ((dw)/dx) . Use this result to differentiate x^(3)(x^(2)+1)(x^(4)+1) w.r.t. x.

If u and v are two differentiable functions of x show that, (d)/(dx)("tan"^(-1)(u)/(v))=(v (du)/(dx)-u(dv)/(dx))/(u^(2)+v^(2)) .

If u and v are two differentiable functions of x and y = uv , prove that , (y')/(y)=(u')/(u)+(v')/(v) where dash (') denotes derivative w.r.t.x.

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 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 y = f(u) is a differentiable functions of u and u = g(x) is a differentiable functions of x such that the composite functions y = f[g(x) ] is a differentiable functions of x then (dy)/(dx) = (dy)/(du) . (du)/(dx) Hence find (dy)/(dx) if y = sin ^(2)x

If u and v are two differentiable functions of x and int uv dx=phi(x)-int[(du)/(dx)intv dx]dx , then the value of phi(x) is-

Prove that (d)/(dx)uv=u(dv)/(dx)+v(du)/(dx) .

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v',v),(u'',v'',2v'):}| .

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v',v),(u'',v'',2v'):}| .