Advertisements
Advertisements
Question
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
Advertisements
Solution
`y = a (cost+logtan(t/2)) and x=asint`
`therefore dy/dt=a[d/dt(cost)+d/dt(log tan (t/2))]=a[-sint+cot(t/2)xxsec^2(t/2)xxt/2]=a[-sint+1/(2sin(t/2)cos(t/2))]`
`dy/dt=a(-sint+1/sint)=a((-sin^2t+1)/sint)=a cos^2t/sint`
`dx/dt=a d/dt(sint)=acost`
`therefore dy/dx=(dy/dt)/(dx/dt)=(a (cos^2t/sint))/acost=cost/sint=cott`
`(d^2y)/(dx^2)=d(cott)/dx=-cosec^2tdt/dx=-cosec^2txx1/(acost)=1/(asin^2tcost)`
APPEARS IN
RELATED QUESTIONS
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.
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 = cos–1 x, find `(d^2y)/dx^2` in terms of y alone.
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`
Find `("d"^2"y")/"dx"^2`, if y = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^-7`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
`sin xy + x/y` = x2 – y
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
Derivative of cot x° with respect to x is ____________.
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))`
