Advertisements
Advertisements
प्रश्न
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
Advertisements
उत्तर
Given that: y = tan–1x
⇒ x = tan y
Differentiating both sides w.r.t. y
`"dx"/"dy"` = sec2y
⇒ `"dy"/'dx" = 1/(sec^2y)` = cos2y
Again differentiating both sides w.r.t. x
⇒ `"d"/"dx"("dy"/"dx") = "d"/"dx"(cos^2y)`
⇒ `("d"^2y)/("dx"^2) = 2cos y * "d"/"dx" (cos y)`
⇒ `("d"^2y)/("dx"^2) = 2cos y(- siny) * "dy"/"dx"`
⇒ `("d"^2y)/("dx"^2) = - 2sin y cos y * cos^2 y`
∴ `("d"^2y)/("dx"^2)` = – 2 sin y cos3y
APPEARS IN
संबंधित प्रश्न
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`
Find the second order derivative of the function.
log x
Find the second order derivative of the function.
ex sin 5x
Find the second order derivative of the function.
e6x cos 3x
Find the second order derivative of the function.
log (log x)
Find the second order derivative of the function.
sin (log x)
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
If y = Aemx + Benx, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`.
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
If x7 . y9 = (x + y)16 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 = `"e"^"x"`
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
sec(x + y) = xy
tan–1(x2 + y2) = a
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
Derivative of cot x° with respect to x is ____________.
If x2 + y2 + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) is ______.
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
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))`
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))`
