Advertisements
Advertisements
प्रश्न
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
Advertisements
उत्तर
y = `log(x + sqrt(x^2 + a^2))^m`
= `mlog(x + sqrt(x^2 + a^2))`
∴ `"dy"/"dx" = m"d"/"dx"[log(x + sqrt(x^2 + a^2))]`
= `m xx (1)/(x + sqrt(x^2 + a^2))."d"/"dx"(x + sqrt(x^2 + a^2))`
= `m/(x + sqrt(x^2 + a^2)) xx [1 + (1)/(2sqrt(x^2 + a^2))."d"/"dx"(x^2 + a^2)]`
= `m/(x + sqrt(x^2 + a^2)) xx [1 + (1)/(2sqrt(x^2 + a^2)).(2x + 0)]`
= `m/(x + sqrt(x^2 + a^2)) xx (sqrt(x^2 + a^2) + x)/(sqrt(x^2 + a^2)`
∴ `"dy"/"dx" = m/sqrt(x^2 + a^2)`
∴ `sqrt(x^2 + a^2)"dy"/"dx"` = m
∴ `(x^2 + a^2)(dy/dx)^2` = m2
Differentiating both sides w.r.t. x, we get
`(x^2 + a^2)."d"/"dx"(dy/dx)^2 + (dy/dx)^2."d"/"dx"(x^2 + a^2) = "d"/"dx"(m^2)`
∴ `(x^2 + a^2) xx 2"dy"/"dx"."d"/"dx"(dy/dx) + (dy/dx)^2 xx (2x + 0)` = 0
∴ `(x^2 + a^2) . 2"dy"/"dx"(d^2y)/(dx^2) + 2x (dy/dx)^2` = 0
Cancelling `2"dy"/"dx"` throughtout, we get
`(x^2 + a^2)(d^2y)/(dx^2) + x"dy"/"dx"` = 0.
संबंधित प्रश्न
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
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
- By using the product rule.
- By expanding the product to obtain a single polynomial.
- By logarithmic differentiation.
Do they all give the same answer?
If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Find `"dy"/"dx"` if y = xx + 5x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Find the second order derivatives of the following : x3.logx
Find the second order derivatives of the following : log(logx)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`2^(cos^(2_x)`
`log (x + sqrt(x^2 + "a"))`
`log [log(logx^5)]`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` 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 y = `x^(x^2)`, then `dy/dx` is equal to ______.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
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.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Evaluate:
`int log x dx`
If xy = yx, then find `dy/dx`
