Advertisements
Advertisements
प्रश्न
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
Advertisements
उत्तर
y = A cos (log x) + B sin (log x) ...(1)
Differentiating both sides w.r.t. x, we get
`"dy"/"dx" = "A""d"/"dx"[cos(logx)] + "B""d"/"dx"[sin(log x)]`
= `"A"[-sin (logx)]."d"/"dx"(logx) + "B"cos(logx)."d"/"dx"(logx)`
= `"A"sin(logx) xx (1)/x "B"cos(logx) xx(1)/x`
∴ `x"d"/"dx"(dy/dx) + "dy"/"dx"."d"/"dx"(x) = -"A""d"/"dx"[sin(logx)] +"B""d"/"dx"[cos(logx)]`
∴ `x(d^2y)/(dx2) + "dy"/"dx" xx 1 = -"A"cos(logx)."d"/"dx"(logx) + "B"[-sin(logx)]."d"/"dx"(logx)`
∴ xy2 + y1 = `-"A"cos(logx) xx(1)/x - "B"sin(logx) xx (1)/x`
∴ x2y2 + xy1 = – [A cos (log x) + B sin (log x)] ...[By (1)]
∴ x2y2 + xy1 + y = 0.
APPEARS IN
संबंधित प्रश्न
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
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.
`(x + 1/x)^x + x^((1+1/x))`
Find `bb(dy/dx)` for the given function:
yx = xy
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Find `"dy"/"dx"` if y = xx + 5x
`"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 x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
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.
Find the second order derivatives of the following : x3.logx
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
Derivative of loge2 (logx) with respect to x is _______.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
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 ______.
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `9^(log_3x)`, find `dy/dx`.
Find the derivative of `y = log x + 1/x` with respect to x.
What is logarithmic differentiation?
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?
What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?
Which derivative correctly represents differentiating \[\ln y\] carefully?
