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

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

(d)/(dx) {x ^(x)}=

(d)/(dx){(x +(1)/(x) )^(3)}=

(d)/(dx) { x ^(1//x)}=

(d)/(dx) {(1)/(sqrt(3x +2))}=

(d)/(dx) {Sec^(-1) e ^(x) }=

(d)/(dx) {cos x ^(0)}=

Let u(x) and v(x) are differentiable function such that (u(x))/(v(x))=7 . If (u'(x))/(v'(x))=p and ((u(x))/(v(x)))^'=q , then (p+q)/(p-q)=

(d)/(dx) {log ((x )/(e ^(tanx)))}=