Advertisements
Advertisements
Question
Find the second order derivative of the function.
e6x cos 3x
Advertisements
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)`
= −3e6x sin 3x + 6e6x cos 3x
= e6x (6 cos 3x − 3 sin 3x)
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)`
= e6x [−6 sin 3x · 3 − 3 cos 3x · 3] + [6 cos 3x − 3 sin 3x] × e6x ⋅ 6
= e6x [−18 sin 3x − 9 cos 3x] + e6x [36 cos 3x − 18 sin 3x]
= e6x [−36 sin 3x + 27 cos 3x]
= 9e6x [3 cos 3x − 4 sin 3x]
APPEARS IN
RELATED QUESTIONS
If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=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.
x2 + 3x + 2
Find the second order derivative of the function.
x20
Find the second order derivative of the function.
x . cos x
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
sin (log x)
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.
If y = cos–1 x, find `(d^2y)/dx^2` in terms of y alone.
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 = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"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 ____________.
Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let,
A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`
and B(x) = [A(x)]T A(x). Then determinant of B(x) ______
If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
Find `(d^2y)/dx^2, "if" y = e^((2x+1))`
