Advertisements
Advertisements
Question
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Solution
We have,
ey ( x +1) = 1
⇒ ey = `1/(x + 1)`
⇒ log `e^y = log (1/(x+1))`
⇒ y = - log (x + 1)
` ⇒ (dy)/(dx) = - 1/ (x + 1) and (d^2 y) /(dx^2) = 1/((x + 1)^2)`
` ⇒ (d^2 y)/(dx^2) = ((dy)/(dx))^2`
APPEARS IN
RELATED QUESTIONS
If `y=log[x+sqrt(x^2+a^2)] ` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
`(log x)^(cos x)`
Differentiate the function with respect to x.
`(x + 1/x)^x + x^((1+1/x))`
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Find `dy/dx` for the function given in the question:
(cos x)y = (cos y)x
Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ′(1).
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w+u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
Evaluate:
`int log x dx`
If xy = yx, then find `dy/dx`
