Advertisements
Advertisements
Question
Find `"dy"/"dx"` if `e^(e^(x - y)) = x/y`
Solution
`e^(e^(x - y)) = x/y`
Taking log on both side
`log e^(e^((x - y))) = log (x/y)`
∴ `e^((x - y)) log e = log x - log y`
∴ ex–y = log x – log y ....[∵ log e = 1]
Differentiating both sides w.r.t. x, we get
`e^(x - y)."d"/"dx"(x - y) = (1)/x - (1)/y"dy"/"dx"`
∴ `e^(x - y)(1 - "dy"/"dx") = (1)/x - (1)/y"dy"/"dx"`
∴ `e^(x - y) - e^(x - y)"dy"/"dx" = (1)/x - (1)/y"dy"/"dx"`
∴ `(1/y - e^(x - y))"dy"/"dx" = (1)/x - e^(x - y)`
`((1 - ye^(x - y))/y)"dy"/"dx" = (1 - xe^(x - y))/x`
∴ `"dy"/"dx" = (y(1 - xe^(x - y)))/((x(1 - ye^(x - y))`.
APPEARS IN
RELATED QUESTIONS
if `y = tan^2(log x^3)`, find `(dy)/(dx)`
Solve : `"dy"/"dx" = 1 - "xy" + "y" - "x"`
Find `"dy"/"dx"`If x3 + x2y + xy2 + y3 = 81
Find `dy/dx if x^2y^2 - tan^-1(sqrt(x^2 + y^2)) = cot^-1(sqrt(x^2 + y^2))`
Find the second order derivatives of the following : e4x. cos 5x
Find `"dy"/"dx"` if, y = (5x3 - 4x2 - 8x)9
Find `"dy"/"dx"` if, y = log(log x)
Find `"dy"/"dx"` if, y = log(10x4 + 5x3 - 3x2 + 2)
Find `"dy"/"dx"` if, y = log(ax2 + bx + c)
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Find `"dy"/"dx"` if, y = `"a"^((1 + log "x"))`
Choose the correct alternative.
If y = (5x3 - 4x2 - 8x)9, then `"dy"/"dx"` =
If y = 2x2 + 22 + a2, then `"dy"/"dx" = ?`
If `"x"^"m"*"y"^"n" = ("x + y")^("m + n")`, then `"dy"/"dx" = "______"/"x"`
Solve the following:
If y = (6x3 - 3x2 - 9x)10, find `"dy"/"dx"`
Find `"dy"/"dx"`, if y = xx.
If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______
If x = cos−1(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______
If y = cos−1 [sin (4x)], find `("d"y)/("d"x)`
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin2x
If y = `1/sqrt(3x^2 - 2x - 1)`, then `("d"y)/("d"x)` = ?
Choose the correct alternative:
If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?
If y = `("e")^((2x + 5))`, then `("d"y)/("d"x)` is ______
If y = x2, then `("d"^2y)/("d"x^2)` is ______
State whether the following statement is True or False:
If y = ex, then `("d"y)/("d"x)` = ex
Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`
If y = `2/(sqrt(a^2 - b^2))tan^-1[sqrt((a - b)/(a + b)) tan x/2], "then" (d^2y)/dx^2|_{x = pi/2}` = ______
If y = `x/"e"^(1 + x)`, then `("d"y)/("d"x)` = ______.
If y = `(cos x)^((cosx)^((cosx))`, then `("d")/("d"x)` = ______.
Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`
If x = a sec3θ and y = a tan3θ, find `("d"y)/("d"x)` at θ = `pi/3`
If f(x) = |cos x|, find f'`((3pi)/4)`
If y = log (cos ex), then `"dy"/"dx"` is:
Differentiate the function from over no 15 to 20 sin (x2 + 5)
y = `sec (tan sqrt(x))`
y = `2sqrt(cotx^2)`
y = `cos sqrt(x)`
Let x(t) = `2sqrt(2) cost sqrt(sin2t)` and y(t) = `2sqrt(2) sint sqrt(sin2t), t ∈ (0, π/2)`. Then `(1 + (dy/dx)^2)/((d^2y)/(dx^2)` at t = `π/4` is equal to ______.
If y = 2x2 + a2 + 22 then `dy/dx` = ______.
Find `"dy"/"dx" if, e ^(5"x"^2- 2"X"+4)`
Find `dy/dx` if, `y=e^(5x^2-2x+4)`
Solve the following:
If y = `root5 ((3x^2 + 8x + 5)^4 ,) "find" "dy"/ "dx"`
Solve the following:
If`y=root(5)((3x^2+8x+5)^4),"find" (dy)/dx`
Find `dy/dx` if, y = `e^(5x^2 - 2x + 4)`
The differential equation of (x - a)2 + y2 = a2 is ______
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
Find `dy/dx` if ,
`x= e^(3t) , y = e^(4t+5)`
lf y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, such that the composite function y = f[g(x)] is a differentiable function of x, then prove that:
`dy/dx = dy/(du) xx (du)/dx`
Hence, find `d/dx[log(x^5 + 4)]`.
Find `dy/dx` if, y = `e^(5x^2-2x+4)`
If y = `tan^-1((6x - 7)/(6 + 7x))`, then `dy/dx` = ______.
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
If y = `root5((3x^2+8x+5)^4)`, find `dy/dx`
Find `dy/dx` if, `y = e^(5x^2 - 2x +4)`
If `y=root5((3x^2+8x+5)^4)`, find `dy/dx`
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10`x + 25x^2`
Find `dy/dx` if, `y = e^(5x^2 - 2x+4)`
Solve the following.
If `y=root(5)((3x^2 + 8x + 5)^4)`, find `dy/dx`
Find `(dy) / (dx)` if, `y = e ^ (5x^2 - 2x + 4)`
If `y = root{5}{(3x^2 + 8x + 5)^4}, "find" dy/dx`.
Find `dy/(dx)` if, y = `e^(5x^2 - 2x + 4)`
If y = `root{5}{(3x^2 + 8x + 5)^4)`, find `(dy)/(dx)`
Find `dy/dx` if, `y = e^(5x^2 - 2x + 4)`.
