Advertisements
Advertisements
प्रश्न
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
Advertisements
उत्तर
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
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
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.
xsin x + (sin x)cos x
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Find `bb(dy/dx)` for the given function:
yx = xy
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
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 .`
Evaluate
`int 1/(16 - 9x^2) dx`
Differentiate : log (1 + x2) w.r.t. cot-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" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
Differentiate 3x w.r.t. logx3.
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 (2x + 3)
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 ______
Derivative of loge2 (logx) with respect to x is _______.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
`d/dx(x^{sinx})` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If y = `x^(x^2)`, then `dy/dx` 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.
If y = `9^(log_3x)`, find `dy/dx`.
Find `dy/dx`, if y = (log x)x.
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
For which type of function is logarithmic differentiation especially useful?
What is \[\frac{1}{y}\cdot\frac{dy}{dx}\] for \[y=\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
For \[y=x^{\sin x}\], \[x>0\], what equation results after taking logarithm on both sides?
After taking logarithms in logarithmic differentiation, which rules are used to simplify products, quotients and powers before differentiation?
What condition must be ensured for an expression inside logarithm?
