Advertisements
Advertisements
प्रश्न
If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`
Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`
Advertisements
उत्तर
Given that
x cos(a+y)=cosy...1
`=>x=(cosy)/cos(a+y)`
Differentiating both sides of the equation (1), we have,
`x xx(-sin(a+y))(dy)/(dx)+1xxcos(a+y)=-siny(dy)/dx`
`=>[siny-xsin(a+y)](dy)/dx=-cos(a+y)`
`=>[siny-cosy/cos(a+y)sin(a+y)]dy/(dx)=-cos(a+y)`
`=>[(cos(a+y)xxsiny-cosysin(a+y))/cos(a+y)]dx/dy=-cos(a+y)`
`=>[sin(a+y-y)]dy/dx=-cos^2(a+y) `
`=>[sina]dy/dx=-cos^2(a+y)`
`=>dy/dx=((-cos^2(a+y))/sina) `
Differentiating once again with respect to x, we have,
`sina(d^2y)/dx^2=-2cos(a+y)sin(a+y)dy/dx`
`=>sina((d^2y)/dx^2)+2cos(a+y)sin(a+y)dy/dx=0`
`=>sina(d^2y)/dx^2+sin2(a+y)dy/dx=0`
Hence proved.
APPEARS IN
संबंधित प्रश्न
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
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.
log x
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
ex sin 5x
Find the second order derivative of the function.
log (log x)
Find the second order derivative of the function.
sin (log x)
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
`sin xy + x/y` = x2 – y
If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^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)`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
Find `(d^2y)/dx^2` if, `y = e^((2x+1))`
Find `(d^2y)/(dx^2) "if", y = e^((2x + 1))`
