Advertisements
Advertisements
Question
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Solution
We have `(x^2 + y^2) = xy`
Differentiating with respect to x, we get
`=> d/dx [(x^2 + y^2)^2] = d/dx (xy)`
`=> 2(x^2 + y^2) d/(dx) (x^2 + y^2) = x (dy)/(dx) + y d/dx (x)`
`=> 2(x^2 + y^2) (2x+ 2y dy/dx) = x (dy/dx) + y (1)`
`=> 4x (x^2 + y^2) + 4y (x^2 + y^2) dy/dx = x dy/dx + y`
`=> 4y(x^2 + y^2) dy/dx - x dy/dx = y - 4x (x^2 + y^2)`
`=> dy/dx [4y(x^2 + y^2) - x] = y - 4x(x^2 + y^2)`
`=> dy/dx = (y - 4x(x^2+y^2))/(4y(x^2 + y^2) - x)`
`=> dy/dx = (4x(x^2 + y^2)-y)/(x-4y(x^2 + y^2))`
APPEARS IN
RELATED QUESTIONS
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `dy/dx` in the following:
xy + y2 = tan x + y
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
Discuss extreme values of the function f(x) = x.logx
Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
Find the nth derivative of the following : cos (3 – 2x)
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
If log(x+y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log (x + y) = log (xy) + a then show that, `dy/dx = (−y^2)/x^ 2`
Find `dy/dx"if", x= e^(3t), y=e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/(dx) "if" , x = e^(3t), y = e^sqrtt`.
If log(x + y) = log(xy) + a, then show that `dy/dx = (-y^2)/x^2`
