If `u=f(x^3),v=g(x^2),f^(prime)(x)=cosx ,a n dg^(prime)(x)=sinx ,t h e n(d u)/(d v)` is `

If y=f(x^3),z=g(x^5),f^(prime)(x)=tanx ,a n dg^(prime)(x)=secx , then find the value of (lim)_(xvec0)(((dy)/(dz)))/x

Let F(x)=f(x)g(x)h(x) for all real x ,w h e r ef(x),g(x),a n dh(x) are differentiable functions. At some point x_0,F^(prime)(x_0)=21 F(x_0),f^(prime)(x_0)=4f(x_0),g^(prime)(x_0)=-7g(x_0), and h^(prime)(x_0)=kh(x_0) . Then k is________

Let f(x)a n dg(x) be two functions which are defined and differentiable for all xgeqx_0dot If f(x_0)=g(x_0)a n df^(prime)(x)>g^(prime)(x) for all x > x_0, then prove that f(x)>g(x) for all x > x_0dot

If g is the inverse function of fa n df^(prime)(x)=sinx ,t h e ng^(prime)(x) is (a) cos e c{g(x)} (b) "sin"{g(x)} (c) -1/("sin"{g(x)}) (d) none of these

Iff(x)=Asin((pix)/2)+b ,f^(prime)(1/2)=sqrt(2)a n d int_0^1f(x)dx=(2A)/pi,t h e ncon s t a n t sAa n dBa r e pi/2a n dpi/2 (b) 2/pia n d3/pi 0a n d-4/pi (d) 4/pia n d0

Given f^(prime)(1)=1"and"d/(dx)(f(2x))=f^(prime)(x)AAx > 0 .If f^(prime)(x) is differentiable then there exies a number c in (2,4) such that f''(c) equals

If f(x)=|x-2|a n dg(x)=f[f(x)],t h e ng^(prime)(x)= ______ for x>2

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

If y=f(a^x)a n df^(prime)(sinx)=(log)_e x ,t h e nfin d(dy)/(dx), if it exists, where pi/2