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.

If vec u , vec v and vec w are three non-coplanar vectors, then prove that ( vec u+ vec v- vec w)dot [( vec u- vec v)xx( vec v- vec w)] = vec u dot (vec v xx vec w)

Differentiate the function w.r.t. x : cosx cdot cos2x cdot cos3x

Differentiate the functions with respect to x: cos x^3 cdot sin^2 (x^5)

Let vec ua n d vec v be unit vectors such that vec uxx vec v+ vec u= vec w and vec wxx vec u= vec vdot Find the value of [ vec u vec v vec w]dot

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'):}|

State true or false for the following statements : (d)/(dx)((u)/(v))=(v(du)/(dx)-u(dv)/(dx))/(v)

Differentiate the function w.r.t. x : (x+3)^2 cdot (x+4)^3 cdot (x+5)^4

Let vec u , vec va n d vec w be such that | vec u|=1,| vec v|=2a n d| vec w|=3. If the projection of vec v along vec u is equal to that of vec w along vec u and vectors vec va n d vec w are perpendicular to each other, then | vec u- vec v+ vec w| equals a 2 b. sqrt(7) c. sqrt(14) d. 14

If f(x) = log_e x, x > 0 , prove that : f (uvw) = f (u) + f (v) +f (w).