Advertisements
Advertisements
प्रश्न
if xx+xy+yx=ab, then find `dy/dx`.
उत्तर १
xx+xy+yx=ab........(i)
`Let u=x^x`
`log u=xlogx`
`1/u*(du)/dx=x * 1/x+logx`
`therefore (du)/dx=x^x(1+logx)`
`Let v=x^y`
`logv =ylogx`
`1/v (dv)/dx=(y/x+logx dy/dx)`
`therefore (dv)/dx=x^y(y/x+logx dy/dx)`
`Let w=y^x`
`logw=x log y`
`1/w.(dw)/dx=(x/y*dy/dx+logy)`
`therefore (dw)/dx=y^x(logy+x/y*dy/dx)`
(i) can be written as
u + v + w = ab
`du/dx+dv/dx+dw/dx=0`
`=>x^x+x^xlogx+x^yy/x+x^y logx dy/dx+y^xlogy+y^x x/y dy/dx=0`
`=>dy/dx(x^ylogx+y^x x/y)=x^x+x^xlogx+x^y y/x+ y^x logy`
`=> dy/dx (x^y*logx+xy^(x-1))=(x^x+x^xlogx+yx^(y-1)+y^x*logy)`
`therefore dy/dx=(x^x+x^xlogx+yx^(y-1)+y^x*logy)/(x^y*logx+xy^(x-1))`
उत्तर २
Let u = xy and v = yx
Then, u + v = ab
Differentiating both sides w.r.t x, we get
संबंधित प्रश्न
Differentiate the function with respect to x.
`(log x)^(cos x)`
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `dy/dx` for the function given in the question:
`xy = e^((x – y))`
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 `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 .`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `(d^2y)/(dx^2)` , if y = log x
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If y = (log x)x + xlog x, find `"dy"/"dx".`
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 x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
The derivative of x2x w.r.t. x is ______.
The derivative of log x with respect to `1/x` is ______.
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
