Advertisements
Advertisements
प्रश्न
Find the nth derivative of the following: log (ax + b)
Advertisements
उत्तर
Let y = log (ax + b)
Then `"dy"/"dx" = "d"/"dx"[log(ax + b)]`
= `(1)/(ax + b)."d"/"dx"(ax + b)`
= `(1)/(ax + b) xx (a xx 1 + 0)`
= `a/"ax + b"`
`(d^2y)/(dx^2) = "d"/"dx"(a/(ax + b))`
= `a"d"/"dx"(ax + b)^-1`
= `a(-1)(ax + b)^-2."d"/"dx"(ax + b)`
= `((-1)a)/((ax + b)^2) xx (a xx 1 + 0)`
= `((-1)a^2)/((ax + b)^2)`
`(d^3y)/(dx^3) = "d"/"dx"[((-1)^1a^2)/(ax + b)^2]`
= `(-1)^1a^2."d"/"dx"(ax + b)^-2`
= `(-1)^1a^2.(-2)(ax + b)^-3."d"/"dx"(ax + b)`
= `((-1)^2. 1.2.a^2)/(ax + b)^3 xx (a xx 1 + 0)`
= `((-1)^2 .2! a^3)/(ax + b)^3`
In general, the nth order derivative is given by
`(d^ny)/(dx^n) = ((-1)^(n - 1).(n - 1)!a^n)/(ax + b)^n`
APPEARS IN
संबंधित प्रश्न
if xx+xy+yx=ab, then find `dy/dx`.
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.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)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 ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `(d^2y)/(dx^2)` , if y = log x
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Find `"dy"/"dx"` if y = xx + 5x
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
Find the second order derivatives of the following : log(logx)
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
Find the nth derivative of the following : log (2x + 3)
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("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 = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/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.
Derivative of loge2 (logx) with respect to x is _______.
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`"d"/"dx" [(cos x)^(log x)]` = ______.
Derivative of `log_6`x with respect 6x to is ______
`2^(cos^(2_x)`
`8^x/x^8`
`log (x + sqrt(x^2 + "a"))`
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `9^(log_3x)`, find `dy/dx`.
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
