Let y=e2x, then d2ydx2d2xdy2 is
1
e-2x
-2e-2x
y=e2x
Find the value of d2ydx2d2xdy2:
Given,y=e2x→(i)
Differentiating equation (i) w.r.t.'x'
dydx=e2x.2
Again differentiating w.r.t.'x'
d2ydx2=2e2x.2=4e2x→(ii)
Differentiating equation (i) w.r.t.'y'
⇒1=2e2xdxdy⇒dxdy=12e2x⇒=12y∵y=e2x
Again differentiating w.r.t.'y'
d2xdy2=12-1y-2=-12e2x-2∵y=e2x=-12e4x→(iii)
Calculate d2ydx2d2xdy2
Substitute (ii) and (iii)
d2ydx2d2xdy2=4e2x×-12e4x=-2e-2x
Hence, the correct answer is option (C)
Let I =∫exe4x+e2x+1dx.J=∫e−xe−4x+e−2x+1dx,Then, for an arbitrary constant c, the value of J-I equals