Advertisements
Advertisements
Question
Find the nth derivative of the following : log (2x + 3)
Advertisements
Solution
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
RELATED QUESTIONS
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
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.
Evaluate
`int 1/(16 - 9x^2) dx`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Find `"dy"/"dx"` if y = xx + 5x
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 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 = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
If f(x) = logx (log x) then f'(e) is ______
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
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)`
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 `log_6`x with respect 6x to is ______
`2^(cos^(2_x)`
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
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)`
Find `dy/dx`, if y = (log x)x.
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
What is logarithmic differentiation?
What is the first step in the standard procedure for \[y=[u(x)]^{v(x)}\]?
Which is the derivative of \[\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
For \[y=x^{\sin x}\], \[x>0\], what equation results after taking logarithm on both sides?
What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?
Which derivative correctly represents differentiating \[\ln y\] carefully?
