Advertisements
Advertisements
प्रश्न
`sin xy + x/y` = x2 – y
Advertisements
उत्तर
Given that: `sin xy + x/y` = x2 – y
Differentiating both sides w.r.t. x
`"d"/"dx" sin(xy) + "d"/"dx"(x/y) = "d"/"dx" (x^2) - "d"/"dx"(y)`
⇒ `cos xy * "d"/"dx" (xy) + (y * "d"/"dx" * x - x * "dy"/"dx")/y^2 = 2x - "dy"/"dx"`
⇒ `cos y [x * "dy"/"dx" + y * 1] + ("y"*1)/"y"^2 - x/y^2 * "dy"/"dx" = 2x - "dy"/"dx"`
⇒ `x cos xy * "dy"/"dx" + y cos xy + 1/y - x/y^2 "dy"/"dx" = 2x - "dy"/"dx"`
⇒ `x cos xy * "dy"/"dx" - x/y^2 * "dy"/"dx" + "dy"/"dx" = -y cos xy - 1/y + 2x`
⇒ `[x cos xy - x/y^2 + 1] "dy"/"dx" = 2x - y cos xy - 1/y`
⇒ `([xy^2 cos xy - x + y^2])/y^2 "dy"/"dx" = (2xy - y^2 cos xy - 1)/y`
⇒ `"dy"/"dx" = (2xy - y^2 cos xy - 1)/y xx y^2/(xy^2 cos xy - x + y^2)`
= `(2xy^2 - y^3 cos(xy) - y)/(xy^2 cos (xy) - x + y^2)`
Hence, `"dy"/"dx" = (2xy^2 - y^3 cos(xy) - y)/(xy^2 cos (xy) - x + y^2)`.
APPEARS IN
संबंधित प्रश्न
If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
x20
Find the second order derivative of the function.
log x
Find the second order derivative of the function.
e6x cos 3x
Find the second order derivative of the function.
tan–1 x
Find the second order derivative of the function.
log (log x)
Find the second order derivative of the function.
sin (log x)
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
sec(x + y) = xy
(x2 + y2)2 = xy
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1
If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
Derivative of cot x° with respect to x is ____________.
Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let,
A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`
and B(x) = [A(x)]T A(x). Then determinant of B(x) ______
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.
If x = a cos t and y = b sin t, then find `(d^2y)/(dx^2)`.
Read the following passage and answer the questions given below:
|
The relation between the height of the plant ('y' in cm) with respect to its exposure to the sunlight is governed by the following equation y = `4x - 1/2 x^2`, where 'x' is the number of days exposed to the sunlight, for x ≤ 3.
|
- Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
- Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/(dx^2)` if, y = `e^((2x+1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, y = `e^(2x +1)`
