Advertisements
Advertisements
Question
`sin xy + x/y` = x2 – y
Advertisements
Solution
Given that: `sin xy + x/y` = x2 – y
Differentiating both sides w.r.t. x
`"d"/"dx" sin(xy) + "d"/"dx"(x/y) = "d"/"dx" (x^2) - "d"/"dx"(y)`
⇒ `cos xy * "d"/"dx" (xy) + (y * "d"/"dx" * x - x * "dy"/"dx")/y^2 = 2x - "dy"/"dx"`
⇒ `cos y [x * "dy"/"dx" + y * 1] + ("y"*1)/"y"^2 - x/y^2 * "dy"/"dx" = 2x - "dy"/"dx"`
⇒ `x cos xy * "dy"/"dx" + y cos xy + 1/y - x/y^2 "dy"/"dx" = 2x - "dy"/"dx"`
⇒ `x cos xy * "dy"/"dx" - x/y^2 * "dy"/"dx" + "dy"/"dx" = -y cos xy - 1/y + 2x`
⇒ `[x cos xy - x/y^2 + 1] "dy"/"dx" = 2x - y cos xy - 1/y`
⇒ `([xy^2 cos xy - x + y^2])/y^2 "dy"/"dx" = (2xy - y^2 cos xy - 1)/y`
⇒ `"dy"/"dx" = (2xy - y^2 cos xy - 1)/y xx y^2/(xy^2 cos xy - x + y^2)`
= `(2xy^2 - y^3 cos(xy) - y)/(xy^2 cos (xy) - x + y^2)`
Hence, `"dy"/"dx" = (2xy^2 - y^3 cos(xy) - y)/(xy^2 cos (xy) - x + y^2)`.
APPEARS IN
RELATED QUESTIONS
Find the second order derivative of the function.
x20
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.
e6x cos 3x
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 = Aemx + Benx, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`.
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
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"`
If `x^3y^5 = (x + y)^8` , 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"`
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
sec(x + y) = xy
tan–1(x2 + y2) = a
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
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.
`"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)`
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))`
Find `(d^2y)/dx^2` if, `y = e^((2x+1))`
Find `(d^2y)/(dx^2) "if", y = e^((2x + 1))`
