Advertisements
Advertisements
Question
if xx+xy+yx=ab, then find `dy/dx`.
Advertisements
Solution 1
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))`
Solution 2
Let u = xy and v = yx
Then, u + v = ab
Differentiating both sides w.r.t x, we get
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
Find `(d^2y)/(dx^2)` , if y = log x
Find `"dy"/"dx"` if y = xx + 5x
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 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 y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
`d/dx(x^{sinx})` = ______
`2^(cos^(2_x)`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
The derivative of x2x w.r.t. x is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
If y = `9^(log_3x)`, find `dy/dx`.
Find the derivative of `y = log x + 1/x` with respect to x.
If xy = yx, then find `dy/dx`
