Advertisements
Advertisements
प्रश्न
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
उत्तर
`xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1
∴ `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1
Differentiating both sides w.r.t. x, we get
`y."d"/"dx"(sqrt(1 - x^2)) + sqrt(1 - x^2)."dy"/"dx" + x."d"/"dx"(sqrt(1 - y^2)) + sqrt(1 - y^2)."d"/"dx"(x)` = 0
∴ `y xx (1)/(2sqrt(1 - x^2))."d"/"dx"(1 - x^2) + sqrt(1 - x^2)."dy"/"dx" + x xx (1)/(2sqrt(1 - y^2))."d"/"dx"(1 - y^2) + sqrt(1 - y^2) xx 1` = 0
∴ `y/(2sqrt(1 - x^2)) xx (0 - 2x) + sqrt(1 - x^2)."dy"/"dx" + x/(2sqrt(1 - y^2)) xx (0 - 2y"dy"/"dx") + sqrt(1 - y^2)` = 0
∴ `(-xy)/sqrt(1 - x^2) + sqrt(1 - x^2)."dy"/"dx" - "xy"/sqrt(1 - y^2)."dy"/"dx" + sqrt(1 - y^2)` = 0
∴ `(sqrt(1 - x^2) - "xy"/sqrt(1 - y^2))"dy"/"dx" = "xy"/sqrt(1 - x^2) - sqrt(1 - y^2)`
∴ `[(sqrt(1 - x^2).sqrt(1 - y^2) - xy)/sqrt(1 - y^2)]"dy"/"dx" = (xy - sqrt(1 - x^2).sqrt(1 - y^2))/sqrt(1 - x^2)`
∴ `(1)/sqrt(1 - y^2)."dy"/"dx" = (-1)/sqrt(1 - x^2)`
∴ `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `dy/dx` in the following:
2x + 3y = sin x
Find `dy/dx` in the following:
ax + by2 = cos y
Find `dy/dx` in the following:
xy + y2 = tan x + y
Find `dy/dx` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
If f (x) = |x − 2| write whether f' (2) exists or not.
Write the derivative of f (x) = |x|3 at x = 0.
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `π/4`.
If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
Find the nth derivative of the following : sin (ax + b)
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following : `(1)/(3x - 5)`
Find the nth derivative of the following : y = eax . cos (bx + c)
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
Choose the correct option from the given alternatives :
If y = sin (2sin–1 x), then dx = ........
Choose the correct option from the given alternatives :
If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
If `x = e^(x/y)`, then show that `"dy"/"dx" = (x - y)/(xlogx)`
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0
Find `"dy"/"dx"` if, yex + xey = 1
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
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"`
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
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`
Find `dy/dx if , x = e^(3t) , y = e^sqrtt`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
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`
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`
