Advertisements
Advertisements
प्रश्न
Find the nth derivative of the following : log (2x + 3)
Advertisements
उत्तर
Let y = log (2x + 3)
Then `"dy"/"dx" = "d"/"dx"[log(2x + 3)]`
= `(1)/(2x + 3)."d"/"dx"(2x + 3)`
= `(1)/(2x + 3) xx (a xx 1 + 0)`
= `a/"2x + 3"`
`(d^2y)/(dx^2) = "d"/"dx"(a/(2x + 3))`
= `a"d"/"dx"(2x + 3)^-1`
= `a(-1)(2x + 3)^-2."d"/"dx"(2x + 3)`
= `((-1)a)/((2x + 3)^2) xx (a xx 1 + 0)`
= `((-1)a)/((2x + 3)^2)`
`(d^3y)/(dx^3) = "d"/"dx"[((-1)^1a^2)/(2x + 3)^2]`
= `(-1)^1a^2."d"/"dx"(2x + 3)^-2`
= `(-1)^1a^2.(-2)(2x + 3)^-3."d"/"dx"(2x + 3)`
= `((-1)^2. 1.2.a^2)/(2x + 3)^3 xx (a xx 1 + 0)`
= `((-1)_^2.2! a^3)/(2x + 3)^3`
In general, the nth order derivative is given by
`(d^ny)/(dx^2) = ((-1)^(n - 1).(n - 1)!2^n)/(2x + 3)^n`.
APPEARS IN
संबंधित प्रश्न
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
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.
`(x + 1/x)^x + x^((1+1/x))`
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Find `bb(dy/dx)` for the given function:
xy + yx = 1
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
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 `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
Differentiate 3x w.r.t. logx3.
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
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.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`d/dx(x^{sinx})` = ______
`log (x + sqrt(x^2 + "a"))`
`log [log(logx^5)]`
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Find `dy/dx`, if y = (log x)x.
If xy = yx, then find `dy/dx`
