Advertisements
Advertisements
Question
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
Advertisements
Solution
Given, ey (x + 1) = 1 ....(1)
Differentiating (1) w.r.t. x, we get
`d/dx e^y (x + 1) = d/dx (1)`
`e^y d/dx (x + 1) + (x + 1) d/dx e^y = 0`
`e^y(1) + (x + 1) e^y dy/dx = 0`
`e^y + (x + 1) e^y dy/dx = 0`
`(x + 1)e^y dy/dx = -e^y`
`(x + 1) dy/dx = -1`
`dy/dx = (-1)/(x + 1)`
Now square both sides:
`(dy/dx)^2 = 1/(x + 1)^2`
Differentiating both sides again with respect to x,
`(d^2 y)/dx^2 = - d/dx 1/(x + 1)`
= `- d/dx (x + 1)^-1`
= `-(-1) (x + 1)^(-1-1)`
= 1 (x + 1)−2
`(d^2 y)/dx^2 = 1/(x + 1)^2`
Hence, `(d^2 y)/dx^2 = (dy/dx)^2`
APPEARS IN
RELATED QUESTIONS
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`
Find the second order derivative of the function.
x . cos x
Find the second order derivative of the function.
log 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"`
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/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 x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
`sin xy + x/y` = x2 – y
sec(x + y) = xy
tan–1(x2 + y2) = a
(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 = 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 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`.
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`
Let \[y=f(x)\]. What is the first derivative of \[y\] with respect to \[x\]?
Which expression defines the second order derivative of \[y\] with respect to \[x\]?
For \[y=\mathrm{A}\sin x+\mathrm{B}\cos x\], what is \[\frac{d^2y}{dx^2}\]?
After differentiating \[\sqrt{1-x^2}\cdot\frac{dy}{dx}=1\], which equation results?
What is \[\frac{d}{dx}\left(\sqrt{1-x^2}\right)\] in the differentiation for \[y=\sin^{-1}x\]?
Differentiating \[(1-x^2)y_{1}^{2}=1\] gives which expression?
Which equation follows from \[(1-x^2)\cdot2y_{1}y_{2}+y_{1}^{2}(0-2x)=0\]?
