Advertisements
Advertisements
Question
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) =((dy)/(dx))^2`
Solution
Given, ey (x + 1) = 1 ...(1)
or, (x + 1) ey = 1
Differentiating both sides with respect to x,
⇒ `1 · ey + (x + 1) ey dy/dx = 0`
`=> e^y + 1 * dy/dx = 0` .... [Putting the value of ey(x + 1) from equation (1)]
`=> dy/dx = - e^y`
Differentiating both sides again with respect to x,
`(d^2 y)/dx^2 = -e^y dy/dx` ...(putting -ey = `dy/dx`)
`= (dy/dx)(dy/dx) = (dy/dx)^2`
Hence, `(d^2 y)/dx^2 = (dy/dx)^2`
APPEARS IN
RELATED QUESTIONS
If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`
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`
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.
x2 + 3x + 2
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.
tan–1 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 = cos–1 x, Find `(d^2y)/dx^2` in terms of y alone.
If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + 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 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 = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^5`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
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"`
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
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))`
