Differentiate

ma ans. is negetive of harry's answer

-32excosx

i mistyped the ans ,ans is 32 e^x cosx...

y=e^xsinx ( given )

dydx= e^xsinx + e^xcosx

d²ydx²=e^xsinx + e^xcosx + e^xcosx - e^xsinx

d²ydx²=2(e^xcosx)

d³ydx³=2(e^xcosx - e^xsinx)

d⁴ydx⁴ =2(e^xcosx - e^xsinx - e^xsinx - e^xcosx)

d⁴y dx⁴= 2(-2e^xsinx)= -4e^xsinx

NOW TAKING -4 AS COMMON WE CAN SEE THAT WE HAVE TO DO AGAIN DERIVATIVE OF e^xsinx ,so AFTER FOURTH DERIVATIVE NEXT FOURTH DERIVATIVE WILL BE ,

d⁸y
dx⁸=-4d⁴xdx⁴

d⁸y
dx⁸=-4(-4e^xsinx)=16e^xsinx

d¹⁰ydx¹⁰=16(d²y/dx²)

d¹⁰ydx¹⁰=16(2e^xcosx)

d¹⁰ydx¹⁰=32 e^xcosx

is it correct now steps seems to be long but when u do roughly, trick works in seconds

hahaha..... no its not so...:P

y = e^xsinx

=> dydx = e^xsinx + e^xcosx = e^x(sinx + cosx) = √2e^xsin (x + Π4)

=> d²yd²x = (√2)²e^x sin (x + 2.Î 4)
.
.
...proceeding this way,

dⁿydxⁿ = (√2)ⁿe^xsin (x + nÎ 4)

so now put n=10 and get the ans 32 e^xcosx :)

yup
euclid explained it better...