Advertisements
Advertisements
प्रश्न
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
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
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.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
(log x)x + xlog x
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`
If `y = e^(acos^(-1)x)`, -1 <= x <= 1 show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
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 (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_10((x^3 - y^3)/(x^3 + y^3)) = 2, "show that" "dy"/"dx" = -(99x^2)/(101y^2)`
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
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.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (2x + 3)
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`d/dx(x^{sinx})` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`2^(cos^(2_x)`
`log [log(logx^5)]`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
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 `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
The derivative of log x with respect to `1/x` is ______.
