Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"` if `e^(e^(x - y)) = x/y`
Advertisements
उत्तर
`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
संबंधित प्रश्न
if `y = tan^2(log x^3)`, find `(dy)/(dx)`
Solve the following differential equation:
x2 dy + (xy + y2) dx = 0, when x = 1 and y = 1
Find `dy/dx if x + sqrt(xy) + y = 1`
Find `"dy"/"dx"` if xey + yex = 1
Find `"dy"/"dx"` if ex+y = cos(x – y)
Find `"dy"/"dx"` if cos (xy) = x + y
Find the second order derivatives of the following : e4x. cos 5x
Find `"dy"/"dx"` if, y = `root(3)("a"^2 + "x"^2)`
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Find `"dy"/"dx"` if, y = `5^(("x" + log"x"))`
Choose the correct alternative.
If y = `sqrt("x" + 1/"x")`, then `"dy"/"dx" = ?`
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x – x2.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`
If y = sec (tan−1x), then `dy/dx` at x = 1 is ______.
If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______
If sin−1(x3 + y3) = a then `("d"y)/("d"x)` = ______
If y = `1/sqrt(3x^2 - 2x - 1)`, then `("d"y)/("d"x)` = ?
Choose the correct alternative:
If y = `x^(sqrt(x))`, then `("d"y)/("d"x)` = ?
If y = `("e")^((2x + 5))`, then `("d"y)/("d"x)` is ______
Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`
y = (6x4 – 5x3 + 2x + 3)6, find `("d"y)/("d"x)`
Solution: Given,
y = (6x4 – 5x3 + 2x + 3)6
Let u = `[6x^4 - 5x^3 + square + 3]`
∴ y = `"u"^square`
∴ `("d"y)/"du"` = 6u6–1
∴ `("d"y)/"du"` = 6( )5
and `"du"/("d"x) = 24x^3 - 15(square) + 2`
By chain rule,
`("d"y)/("d"x) = ("d"y)/square xx square/("d"x)`
∴ `("d"y)/("d"x) = 6(6x^4 - 5x^3 + 2x + 3)^square xx (24x^3 - 15x^2 + square)`
If y = (sin x2)2, then `("d"y)/("d"x)` is equal to ______.
`"d"/("d"x) [sin(1 - x^2)]^2` = ______.
Given f(x) = `1/(x - 1)`. Find the points of discontinuity of the composite function y = f[f(x)]
If f(x) = |cos x|, find f'`((3pi)/4)`
If f(x) = |cos x – sinx|, find `"f'"(pi/6)`
If `sqrt(1 - x^2) + sqrt(1 - y^2) = a(x - y)`, prove that `(dy)/(dx) = sqrt((1 - y^2)/(1 - x^2))`.
y = `cos sqrt(x)`
Let f(x) = log x + x3 and let g(x) be the inverse of f(x), then |64g"(1)| is equal to ______.
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 ______.
Let f(x) = x | x | and g(x) = sin x
Statement I gof is differentiable at x = 0 and its derivative is continuous at that point.
Statement II gof is twice differentiable at x = 0.
Find `dy/dx` if, `y=e^(5x^2-2x+4)`
Find `"dy"/"dx"` if, `"y" = "e"^(5"x"^2 - 2"x" + 4)`
The differential equation of (x - a)2 + y2 = a2 is ______
Find the rate of change of demand (x) of acommodity with respect to its price (y) if
`y = 12 + 10x + 25x^2`
Find `dy/dx` if ,
`x= e^(3t) , y = e^(4t+5)`
Solve the following:
If y = `root5((3x^2 +8x+5)^4`,find `dy/dx`
If y = `log((x + sqrt(x^2 + a^2))/(sqrt(x^2 + a^2) - x))`, find `dy/dx`.
Find `dy/dx` if, y = `e^(5x^2 -2x + 4)`
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 = `root{5}{(3x^2 + 8x + 5)^4)`, find `(dy)/(dx)`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/(dx)`.
