Advertisements
Advertisements
Question
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
Solution
x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ)`
∴ `x/a = sqrt(secθ - tanθ), y/a = sqrt(secθ + tanθ)`
∴ `sec θ - tanθ = x^2/a^2` ...(1)
`sec θ + tanθ = y^2/a^2` ...(2)
Adding (1) and (2), we get
2secθ = `x^2/a^2 + y^2/a^2`
= `(x^2 + y^2)/a^2`
∴ secθ = `(x^2 + y^2)/(2a^2)`
Subtracting (1) from (2), we get
2tanθ = `y^2/a^2 - x^2/a^2`
= `(y^2 - x^2)/a^2`
∴ tanθ = `(y^2 - x^2)/(2a^2)`
∴ sec2θ - tan2θ = 1 gives,
`((x^2 + y^2)/(2a^2))^2 - ((y^2 - x^2)/(2a^2))^2` = 1
∴ (x2 + y2)2 - (y2 - x2)2 = 4a4
∴ (x4 + 2x2y2 + y4) - (y4 - 2x2y2 + x4) = 4a4
∴ 4x2y2 = 4a4
∴ x2y2 = a4
Differentiating both sides w.r.t. x, we get
`x^2."d"/"dx"(y^2) + y^2."d"/"dx"(x^2)` = 0
∴ `x^2 xx 2y"dy"/"dx" + y^2 xx 2x` = 0
∴ `2x^2y"dy"/"dx"` = -2xy2
∴ `"dy"/"dx" = -y/x`.
APPEARS IN
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
If `y=log[x+sqrt(x^2+a^2)] ` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
`(log x)^(cos x)`
Differentiate the function with respect to x.
`x^x - 2^(sin x)`
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Find `dy/dx` for the function given in the question:
`xy = e^((x – y))`
Differentiate w.r.t. x the function:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Find the second order derivatives of the following : log(logx)
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
Find the nth derivative of the following : log (ax + b)
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If f(x) = logx (log x) then f'(e) is ______
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`log [log(logx^5)]`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
