Advertisements
Advertisements
Question
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
Advertisements
Solution
x2 + 6xy + y2 = 10 ...(i)
Differentiating both sides w.r.t. x, we get
`2x + 6(x ("d"y)/("d"x) + y) + 2y ("d"y)/("d"x)` = 0
∴ `2x + 6x ("d"y)/("d"x) + 6y + 2y ("d"y)/("d"x) = 0`
∴ `(2x + 6y) + (6x + 2y) ("d"y)/("d"x) = 0`
∴ `("d"y)/("d"x) = - (x + 3y)/(3x + y)` ...(ii)
∴ (3x + y) `("d"y)/("d"x)` = − (x + 3y)
Again, differentiating both sides w.r.t. x, we get
`(3x + y) ("d"^2y)/("d"x^2) + ("d"y)/("d"x) (3 + ("d"y)/("d"x)) = - (1 + 3 * ("d"y)/("d"x))`
∴ `3 ("d"y)/("d"x) + (("d"y)/("d"x))^2 + 1 + 3("d"y)/("d"x) = - ("d"^2y)/("d"x^2)`(y + 3x)
∴ `(("d"y)/("d"x))^2 + 6 ("d"y)/("d"x) + 1 = - ("d"^2y)/("d"x^2)`(y + 3x)
∴ `[- ((x + 3y)/(3x + y))]^2 + 6 [(- (x + 3y))/(3x + y)] + 1`
= `-("d"^2y)/("d"x^2)` (y + 3x) ...[From (ii)]
By solving, we get
`(x^2 + 9y^2 + 6xy - 6xy - 18x^2 - 18y^2 - 54xy + y^2 + 9x^2 + 6xy)/(y + 3x)^2 = - ("d"^2y)/("d"x^2)`(y + 3x)
∴ `- ("d"^2y)/("d"x^2) (y + 3x)^3 = - 8x^2 - 8y^2 - 48xy`
= `-8 (x^2 + y^2 + 6xy)`
= −8 × 10 ...[from (i)]
= −80
∴ `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
APPEARS IN
RELATED QUESTIONS
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`
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.
tan–1 x
Find the second order derivative of the function.
log (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 = Aemx + Benx, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`.
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"x"`
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
`sin xy + x/y` = x2 – y
sec(x + y) = xy
tan–1(x2 + y2) = a
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
If x2 + y2 + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) 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 ______.
If x = a cos t and y = b sin t, then find `(d^2y)/(dx^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))`
