Advertisements
Advertisements
Question
If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`
Advertisements
Solution
Given that ex + ey = ex+y.
Differentiating both sides w.r.t. x, we have
`"e"^x + "e"^y ("d"y)/("d"x) = "e"^(x + y) (1 + ("d"y)/("d"x))`
or `("e"^y - "e"^(x + y)) ("d"y)/("d"x) = "e"^(x + y) - "e"^x`
Which implies that `("d"y)/("d"x) = ("e"^(x + y) - "e"^x)/("e"^y - "e"^(x + y))`
= `("e"^x + "e"^y - "e"^x)/("e"^y - "e"^x - "e"^y)`
= –ey–x.
APPEARS IN
RELATED QUESTIONS
if `y = tan^2(log x^3)`, find `(dy)/(dx)`
Find `"dy"/"dx"`If x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if cos (xy) = x + y
Find the second order derivatives of the following : e2x . tan x
Find `"dy"/"dx"` if, y = `root(3)("a"^2 + "x"^2)`
Choose the correct alternative.
If y = `sqrt("x" + 1/"x")`, then `"dy"/"dx" = ?`
Fill in the Blank.
If 3x2y + 3xy2 = 0, then `(dy)/(dx)` = ______.
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)` = ?
State whether the following statement is True or False:
If y = ex, then `("d"y)/("d"x)` = ex
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 u = x2 + y2 and x = s + 3t, y = 2s - t, then `(d^2u)/(ds^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 y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.
If `sqrt(1 - x^2) + sqrt(1 - y^2) = a(x - y)`, prove that `(dy)/(dx) = sqrt((1 - y^2)/(1 - x^2))`.
y = `2sqrt(cotx^2)`
If `d/dx` [f(x)] = ax+ b and f(0) = 0, then f(x) is equal to ______.
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, y = `e^(5x^2-2x+4)`
If x = Φ(t) is a differentiable function of t, then prove that:
`int f(x)dx = int f[Φ(t)]*Φ^'(t)dt`
Hence, find `int(logx)^n/x dx`.
If y = `sqrt((1 - x)/(1 + x))`, then `(1 - x^2) dy/dx + y` = ______.
Find `dy/dx` if, y = `e^(5x^2 -2x + 4)`
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"^(5"x"^2 - 2"x" + 4)`
If `y = root{5}{(3x^2 + 8x + 5)^4}, "find" dy/dx`.
Find `dy/(dx)` if, y = `e^(5x^2 - 2x + 4)`
