Advertisements
Advertisements
Question
Find the second order derivative of the function.
e6x cos 3x
Solution
Let, y = e6x cos 3x
Differentiating both sides with respect to x,
`dy/dx = e^(6x) d/dx cos 3 x + cos 3 x d/dx e^(6x)`
`= e^(6x) (- sin 3 x) d/dx (3x) + cos 3 x * e^(6x) d/dx (6x)`
`= - 3 e^(6x) sin 3 x + 6 e^(6x) cos 3 x`
`= e^(6x) (6 cos 3 x - 3 sin 3 x)`
Differentiating both sides again with respect to x,
`(d^2 y)/dx^2 = e^(6x) d/dx (6 cos 3 x - 3 sin 3 x) + (6 cos 3 x - 3 sin 3 x) d/dx e^(6x)`
`= e^(6x) [6 (- sin 3x) d/dx (3x) - 3 cos 3x d/dx (3x)] + [6 cos 3x - 3 sin 3x]e^(6x) d/dx (6x)`
`= e^(6x) [-6 sin 3x * 3 - 3 cos 3x . 3] + [6 cos 3x - 3 sin 3x] xx e^(6x) * 6`
`= e^(6x) [- 18 sin 3x - 9 cos 3 x] + e^(6x) [36 cos 3x - 18 sin 3x]`
`= e^(6x) [- 36 sin 3x + 27 cos 3x]`
`= 9 e^(6x) [3 cos 3x - 4 sin 3x]`
APPEARS IN
RELATED QUESTIONS
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`
Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
`x^20`
Find the second order derivative of the function.
x . cos x
Find the second order derivative of the function.
log x
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
ex sin 5x
Find the second order derivative of the function.
tan–1 x
Find the second order derivative of the function.
sin (log x)
If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) =((dy)/(dx))^2`
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
tan–1(x2 + y2) = a
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
Derivative of cot x° with respect to x is ____________.
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2 if, y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, y = `e^((2x + 1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, y = `e^(2x +1)`
