Find the derivative with respect to x
(1) y=sin√x2+a2
let y=sinu
dydu=cosu
and u=√v dudv=12√v
v=x2+a2 dvdx=2x
dydx=dydu ⋅dudv⋅dvdx
=cosu⋅12√v⋅2x
=xcos√x2+a2√x2+a2 Ans
(2) y=log2(sinx3)
=loge(sinx3)×log2e
let y=logeu×log2e
dydu=log2e⋅1u
u=sinv dudv=cosv
v=x3 dvdx=3x2
dydx=dydu⋅ dudv⋅dvdx
=log2e⋅1u⋅cosv⋅3x2
=3x2cot(x3)log2e Ans
(3) y=e√cosx
let, y=eu ∴y=eu
dydu=eu
u=√v dudv=12√v
v=cosx dvdx=−sinx
dydx=dydu⋅ dudv⋅dvdx
=eu ⋅12√v⋅(−sinx)
=−e√cosxsinx2√cosx Ans
(4) y=1√log(secx)
let, y=1u dydu=−1u2
u=√v dudv=12√v
v=log(secx) dvdx=tanx
dydx=dydu⋅ dudv⋅dvdx
=−1u2⋅12√v⋅tanx
=−tanx2log(secx)√log(secx)
=−12tanx[log(secx)]3/2 Ans
(5) y=cos−1√2x−3
let y=cos−1u dydu=−1√1−u2
u=√v dudv=12√v
v=2x−3 dvdx=2
dydx=dydu⋅ dudv⋅dvdx
=−1√1−u2⋅12√v⋅2
=1√1−2x+3⋅1√2x−3
=−1√4−2x√2x−3
=−1√2⋅1√2−x√2x−3 Ans
(6) y=xx2+ax2
let, y=u+v dydx=dudx+dvdx
u=xx2
logu=x2logx
1ududx=2xlogx+x
dudx=xx2[2xlogx+x]
v=ax2
dvdx=2xax2logea
∴dydx=xx2[2xlogx+x]+2xax2logea Ans
(7) y=(sinx)cosx+e3x
let, y=u+v
dydx=dudx+dvdx
u=(sinx)cosx
logu=cosxlogsinx
1ududx=−sinxlog(sinx)+cosx⋅cotx
dudx=(sinx)cosx[cosx⋅cotx−sinxlog(sinx)]
dvdx=3e3x
∴dydx=(sinx)cosx[cosx⋅cotx−sinxlog(sinx)]+3e3x Ans
(8) y=xx+(sinx)x
let, y=u+v
dydx=dudx+dvdx
u=xx
logu=xlogx
1ududx=logx+1
dudx=xx[logx+1]
v=(sinx)x
logv=xlog(sinx)
1vdvdx=log(sinx)+xcotx
dvdx=(sinx)x[log(sinx)+xcotx]
∴dydx=xx[logx+1]+(sinx)x[log(sinx)+xcotx] Ans
(9) y=(tanx)cotx+(cotx)tanx
let y=u+v
dydx=dudx+dvdx
u=(tanx)cotx
logu=cotxlog(tanx)
⟹1u⋅dudx=−cosec2xlog(tanx)+sec2xtan2x
=(tanx)cotx[−cosec2xlog(tanx)+sec2xtan2x]
v=(cotx)tanx
logv=tanxlog(cotx)
⟹1v⋅dvdx=sec2xlog(cotx)−cosec2xcot2x
dvdx=(cotx)tanx[sec2xlog(cotx)−cosec2xcot2x]
∴dydx=(tanx)cotx[−cosec2xlog(tanx)+sec2xtan2x]
+(cotx)tanx[sec2xlog(cotx)−cosec2xcot2x] Ans
(10) y=(sinx)cosx+(cosx)sinx
let y=u+v
dydx=dudx+dvdx
u=(sinx)cosx
logu=cosxlog(sinx)
⟹1u⋅dudx=−sinxlog(sinx)+cosxcotx
dudx=(sinx)cosx[−sinxlog(sinx)+cosxcotx]
v=(cosx)sinx
logv=sinxlog(cosx)
⟹1v⋅dvdx=cosxlog(cosx)−secxtanx
dvdx=(cosx)sinx[cosxlog(cosx)−secxtanx]
∴dydx=(sinx)cosx[−sinxlog(sinx)+cosxcotx]
+(cosx)sinx[cosxlog(cosx)−secxtanx] Ans
(11) y=ecos−1x+x√x
let y=u+v
dydx=dudx+dvdx
u=ecos−1x
⟹dudx=−ecos−1x√1−x2
v=x√x
logv=√x⋅logx
⟹1v⋅dvdx=12√x⋅logx+1√x
dvdx=x√x[12√x⋅logx+1√x]
∴dydx=−ecos−1x√1−x2
+x√x[12√x⋅logx+1√x] Ans
(12) y=(sinx)tanx+(cosx)secx
let y=u+v
dydx=dudx+dvdx
u=(sinx)tanx
logu=tanxlog(sinx)
⇒1u⋅dudx=sec2xlog(sinx)+1
dudx=(sinx)tanx[sec2xlog(sinx)+1]
v=(cosx)secx
logv=secxlog(cosx)
⇒1v⋅dvdx=secxtanxlog(cosx)−secxtanx
dvdx=(cosx)secx[secxtanxlog(cosx)−secxtanx]
∴dydx=(sinx)tanx[sec2xlog(sinx)+1]
+(cosx)secx[secxtanxlog(cosx)−secxtanx] Ans
(13) y=xsinx+(sinx)cosx
let y=u+v
dydx=dudx+dvdx
u=xsinx
logu=sinxlogx
⇒1u⋅dudx=cosxlogx+sinxx
dudx=xsinx[cosxlogx+sinxx]
v=(sinx)cosx
logv=cosxlog(sinx)
⟹1v⋅dvdx=−sinxlog(sinx)+cosxcotx
dvdx=(sinx)cosx[−sinxlog(sinx)+cosxcotx]
∴dydx=xsinx[cosxlogx+sinxx]
+(sinx)cosx[−sinxlog(sinx)+cosxcotx] Ans
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