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
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
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.
`(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)`
Find `bb(dy/dx)` for the given function:
yx = xy
Find `bb(dy/dx)` for the given function:
(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 `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
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 y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
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`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.logx
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
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 = 5x. x5. xx. 55 , 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.
Derivative of loge2 (logx) with respect to x is _______.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
Derivative of `log_6`x with respect 6x to is ______
`2^(cos^(2_x)`
`8^x/x^8`
`log [log(logx^5)]`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
After differentiating \[\log y=v(x)\cdot\log[u(x)]\], which equation is obtained?
What is \[\frac{1}{y}\cdot\frac{dy}{dx}\] for \[y=\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
After taking logarithms in logarithmic differentiation, which rules are used to simplify products, quotients and powers before differentiation?
Which derivative correctly represents differentiating \[\ln y\] carefully?
