Advertisements
Advertisements
प्रश्न
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
Advertisements
उत्तर
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = my.
Explanation:
y = `("x" + sqrt("x"^2 - 1))^"m"`
Differentiating both sides w.r.t. x, we get
`"dy"/"dx" = "m" ("x" + sqrt("x"^2 - 1))^"m - 1" * "d"/"dx" ("x" + sqrt("x"^2 - 1))`
`= "m" ("x" + sqrt("x"^2 - 1))^"m"/("x" + sqrt("x"^2 - 1))^1 * [1 + 1/(2sqrt("x"^2 - 1)) * "d"/"dx" ("x"^2 - 1)]`
`= "my"/("x" + sqrt("x"^2 - 1)) xx [(1 + 1/(2sqrt("x"^2 - 1))) ("2x")]`
`= "my"/("x" + sqrt("x"^2 - 1)) xx (1 + "x"/sqrt("x"^2 - 1))`
∴ `"dy"/"dx" = "my"/("x" + sqrt("x"^2 - 1)) xx (sqrt("x"^2 - 1) + "x")/sqrt("x"^2 - 1)`
∴ `"dy"/"dx" = "my"/sqrt("x"^2 - 1)`
∴ `sqrt("x"^2 - 1) * "dy"/"dx" = "my"`
APPEARS IN
संबंधित प्रश्न
Find `bb(dy/dx)` in the following:
ax + by2 = cos y
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Write the derivative of f (x) = |x|3 at x = 0.
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
Differentiate tan-1 (cot 2x) w.r.t.x.
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 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`.
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following : y = eax . cos (bx + c)
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 ...
If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
Find `"dy"/"dx"` if, xy = log (xy)
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 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`
