Advertisements
Advertisements
प्रश्न
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
Advertisements
उत्तर
`"e"^"x" + "e"^"y" = "e"^("x + y")` .....(i)
Differentiating both sides w.r.t.x, we get,
`"d"/"dx" "e"^"x" + "d"/"dx" "e"^"y" = "d"/"dx" "e"^("x + y")`
`"e"^"x" "d"/"dx" "x" + "e"^"y" "d"/"dx" "y" = "e"^("x + y") "d"/"dx" ("x + y") ...("d"/"dx" "e"^"x" = "e"^"x")`
`"e"^"x". (1) + "e"^"y" "dy"/"dx" = "e"^("x + y"). ["d"/"dx" "x" + "d"/"dx" "y"] ...("d"/"dx" "x" = 1)`
∴ `"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"`
∴ `["e"^"y" − ("e"^"x" + "e"^"y")] "dy"/"dx" = ("e"^"x" + "e"^"y") − "e"^"x" ...["From (i)"]`
∴ `("e"^"y" - "e"^"x" - "e"^"y") "dy"/"dx" = ("e"^"x" + "e"^"y" - "e"^"x")`
∴ `(- "e"^"x") "dy"/"dx" = ("e"^"y")`
∴ `"dy"/"dx" = - ("e"^"y")/("e"^"x")`
∴ `"dy"/"dx" = - "e"^("y - x")`
APPEARS IN
संबंधित प्रश्न
Find `bb(dy/dx)` in the following:
xy + y2 = tan x + y
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
Find `bb(dy/dx)` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Differentiate xx w.r.t. xsix.
If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.
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:
`(1)/x`
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"` = ...........
Choose the correct option from the given alternatives :
If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1:
| x | f(x) | g(x) | f')x) | g'(x) |
| 0 | 1 | 5 | `(1)/(3)` | |
| 1 | 3 | – 4 | `-(1)/(3)` | `-(8)/(3)` |
(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...
Differentiate the following w.r.t. x:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`
If x5· y7 = (x + y)12 then show that, `dy/dx = y/x`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
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`.
