Advertisements
Advertisements
प्रश्न
Find `bb(dy/dx)` for the given function:
yx = xy
Advertisements
उत्तर
Given, yx = xy
Taking logarithm of both sides,
log yx = log xy
x log y = y log x ...(i)
Differentiating (i) w.r.t. x,
`x d/dx log y + log y d/dx (x)= y d/dx log x + log x d/dx y`
`=> x xx 1/y dy/dx + log y xx 1 = y xx 1/x + log x dy/dx`
`=> x/y dy/dx + log y = y/x + log x dy/dx`
`=> x/y dy/dx - log x dy/dx = y /x - log y`
`=> dy/dx (x/y - log x) = y /x - log y` ...(ii)
On multiplying both sides of (ii) by xy, we get
`=> dy/dx (x^2 - xy log x) = y ^2 - xy log y`
`therefore dy/dx = (y ^2 - xy log y)/(x^2 - xy log x)`
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
`(x + 1/x)^x + x^((1+1/x))`
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `"dy"/"dx"` if y = xx + 5x
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
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 = `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 = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.logx
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 y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
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})` = ______
Derivative of `log_6`x with respect 6x to is ______
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
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 ______.
Find `dy/dx`, if y = (log x)x.
Evaluate:
`int log x dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
