If `u,v and w` be any three derivable functions of `x`,
obtain a formula for `d/(dx)(u upsilonw)`.

If u, v and w are functions of x, then show that d/dx(u cdot v cdot w) = du/dx (v cdotw + u cdot dv/dx cdot w +u cdot v dw/dx) in two ways - first by repeated application of product rule, second by logarithmic differentiation.

Find the derivative of the function sqrt(a+sqrt(a+x) w.r.t.x

Let u(x) and v(x) be differentiable functions such that (u(x))/(v(x))=7 If (u^(prime)(x))/(v^(prime)(x))=p and ((u(x))/(v(x)))^'=q ,then (p+q)/(p-q) has the value (a) 1 (b) 0 (c) 7 (d) -7

Find the derivatives of the following functions at any point of their domains: log |sin x|

Differentiate the following functions w.r.t.x : x log x +log (log x)

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

Let u(x) and v(x) be two continous functions satisfying the differential equations (du)(dx) +p(x)u=f(x) and (dv)/(dx)+p(x)v=g(x) , respectively. If u(x_(1)) gt v(x_(1)) for some x_(1) and f(x) gt g(x) for all x gt x_(1) , prove that any point (x,y) ,where x gt x_(1) , does not satisfy the equations y=u(x) and y=v(x) simultaneously.

Find the derivatives of the following functions at any point of their domains: e^(x^2)