Advertisements
Advertisements
Question
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
Advertisements
Solution
Given that: xm . yn = (x + y)m+n
Taking log on both sides
log xm . yn = log (x + y)m+n ......[∵ log xy = log x + log y]
⇒ log xm + log yn = (m + n) log (x + y)
⇒ m log x + n log y = (m + n) log (x + y)
Differentiating both sides w.r.t. x
⇒ `"m" * "d"/"dx" log x + "n" * "d"/"dx" log y = ("m" + "n") "d"/"dx" log (x + y)`
⇒ `"m" * 1/x + "n" * 1/y * "dy"/"dx" = ("m" + "n") * 1/(x + y) (1 + "dy"/"dx")`
⇒ `"m"/x + "n"/y * "dy"/"dx" = ("m" + "n")/(x + y) * (1 + "dy"/"dx")`
⇒ `"m"/x + "n"/y * "dy"/"dx" = ("m" + "n")/(x + y) + ("m" + "n")/(x + y) * "dy"/"dx"`
⇒ `"n"/y * "dy"/"dx" - ("m" + "n")/(x + y) * "dy"/"dx" = ("m" + "n")/(x + y) - "m"/x`
⇒ `("n"/y - ("m" + "n")/(x + y))"dy"/"dx" = ("m" + "n")/(x + y) - "m"/x`
⇒ `(("n"x + "n"y - "m"y - "n"y)/(y(x + y)))"dy"/"dx" = (("m"x + "n"x - "m"x - "m"y)/(x(x + y)))`
⇒ `(("n"x - "m"y)/(y(x + y))) "dy"/"dx" = (("n"x- "m"y)/(x(x + y)))`
⇒ `"dy"/"dx" = ("n"x - "m"y)/(x(x + y)) xx (y(x + y))/("n"x - "m"y)`
⇒ `"dy"/"dx" = y/x`
Hence proved.
APPEARS IN
RELATED QUESTIONS
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.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
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
Differentiate the function with respect to x.
xsin x + (sin x)cos 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 = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `dy/dx` if y = xx + 5x
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
Find the second order derivatives of the following : log(logx)
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
If f(x) = logx (log x) then f'(e) is ______
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 log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`d/dx(x^{sinx})` = ______
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`log (x + sqrt(x^2 + "a"))`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
The derivative of x2x w.r.t. x is ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
If xy = yx, then find `dy/dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
