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 xx+xy+yx=ab, then find `dy/dx`.
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.
xx − 2sin x
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
xsin x + (sin x)cos 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 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 cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
Find `dy/dx` if y = xx + 5x
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Find `"dy"/"dx"` if y = xx + 5x
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 = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
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)`.
Differentiate 3x w.r.t. logx3.
Find the nth derivative of the following: log (ax + b)
Find the nth derivative of the following : log (2x + 3)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
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 ______.
The derivative of x2x w.r.t. x is ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
If y = `9^(log_3x)`, find `dy/dx`.
Find the derivative of `y = log x + 1/x` with respect to x.
If xy = yx, then find `dy/dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
