Advertisements
Advertisements
Question
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Advertisements
Solution
ex + ey = ex+y ...[1]
Differentiating both sides w.r.t. x, we get
`e^x + e^y."dy"/"dx" = e^(x + y)."d"/"dx"(x + y)`
∴ `e^x + e^y."dy"/"dx" = e^(x + y).(1 + "dy"/"dx")`
∴ `e^x + e^y"dy"/"dx" = e^(x + y) + e^(x + y)"dy"/"dx"`
∴ `(e^y - e^(x + y))"dy"/"dx" = e^(x + y) –e^x`
∴ `"dy"/"dx" = (e^(x +y) - e^x)/(e^y - e^(x + y)`
= `((e^x + e^y) - e^x)/(e^y - (e^x + e^y))` ...[Using (1), since `e^(x + y) = e^x + e^y`]
= `e^y/-e^x`
= – ey – x
RELATED QUESTIONS
If y=eax ,show that `xdy/dx=ylogy`
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
ax + by2 = cos y
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
Find `bb(dy/dx)` in the following:
sin2 y + cos xy = k
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
if `x^y + y^x = a^b`then Find `dy/dx`
Show that the derivative of the function f given by
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Is |sin x| differentiable? What about cos |x|?
If f (x) = |x − 2| write whether f' (2) exists or not.
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `dy/dx if x^3 + y^2 + xy = 7`
Differentiate e4x + 5 w.r..t.e3x
Find `"dy"/"dx"` if x = at2, y = 2at.
Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Differentiate xx w.r.t. xsix.
Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `π/4`.
If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : sin (ax + b)
Find the nth derivative of the following : y = eax . cos (bx + c)
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
Choose the correct option from the given alternatives :
If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "then" "dy"/"dx"` = ...........
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, then `"dy"/"dx"` = .........
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
If x = `e^(x/y)`, then show that `dy/dx = (x - y)/(xlogx)`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
y = `e^(x3)`
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
`"If" log(x+y) = log(xy)+a "then show that", dy/dx=(-y^2)/x^2`
If log(x+y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log (x + y) = log (xy) + a then show that, `dy/dx = (−y^2)/x^ 2`
Find `dy/dx if , x = e^(3t) , y = e^sqrtt`
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Solve the following.
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if, x = e3t, y = `e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log(x + y) = log(xy) + a then show that, `dy/dx = (−y^2)/x^2`
Find `dy/(dx) "if" , x = e^(3t), y = e^sqrtt`.
