हिंदी

Find the nth derivative of the following : (ax + b)m - Mathematics and Statistics

Advertisements
Advertisements

प्रश्न

Find the nth derivative of the following : (ax + b)m 

योग
Advertisements

उत्तर

Let y = (ax + b)m

Then `"dy"/"dx" = "d"/"dx"(ax + b)^m`

= `m(ax + b)^(m-1)."d"/"dx"(ax + b)`
= m(ax + b)m–1 x (a x 1 + 0)
= am(ax + b)m–1 

`(d^2y)/(dx^2) = "d"/"dx"[am(ax + b)^(m-1)]`

= `am"d"/"dx"(ax + b)^(m - 1)`

= `am(m - 1)(ax + b)^(m - 2)."d"/"dx"(ax + b)`

= am(m – 1)(ax + b)m–2  x (a x 1 + 0)
= a2m(m –1) (ax + b)m–2

`(d^2y)/(dx^3) = "d"/"dx"[a^2m(m - 1)"d"/"dx"(ax + b)^(m - 2)]`

= `a^2m(m - 1)"d"/"dx"(ax + b)^(m - 2)`

= `a^m(m - 1)(m - 2)(ax + b)^(m - 3)"d"/"dx"(ax + b)`

= a2m(m – 1)(m – 2)(ax + b)m–3 x (a x 1 + 0)
= a3m(m – 1)(m – 2)(ax + b)m–3
In general, the nth order derivative is given by 
`(d^ny)/(dx^n) = a^nm(m - 1)(m - 2)` ...(m – n + 1)(ax + b)m–n

Case (i) : if m > 0, m > n, then

`(d^ny)/(dx^n) = ((a^n.m(m - 1)(m - 2)...(m - n + 1)(m - n)...3.2.1))/((m - n)(m - n - 1)...3.2.1) xx (ax + b)^(m - n)`

∴ `(d^2y)/(dx^n) = ((a^n.m!(ax + b)^(m - n)))/((m - n)!`,If m > 0, m > n.

Case (ii) : if m > 0 and m < n, then its mth order derivative is a constant and every derivatives after mth order are zero.

∴ `(d^ny)/(dx^n)` = 0, if m > 0, m = n.

Case (iii) : If m > 0, m = n, then

`(d^ny)/(dx^n) = a^n . n(n - 1)(n - 2)...(n - n + 1)(ax + b)^(n - n)`

= an.n(n – 1)(n – 2) ... 1.(ax + b)0

∴ `(d^ny)/(dx^n)` = an . n!, if m > 0, m = n.

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 1: Differentiation - Exercise 1.5 [पृष्ठ ६०]

APPEARS IN

बालभारती Mathematics and Statistics 2 (Arts and Science) [English] Standard 12 Maharashtra State Board
अध्याय 1 Differentiation
Exercise 1.5 | Q 4.01 | पृष्ठ ६०

वीडियो ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्न

Find `bb(dy/dx)` in the following:

ax + by2 = cos y


Find `bb(dy/dx)` in the following:

xy + y2 = tan x + y


Find `bb(dy/dx)` in the following:

`y = sin^(-1)((2x)/(1+x^2))`


if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`


Show that the derivative of the function f given by 

\[f\left( x \right) = 2 x^3 - 9 x^2 + 12x + 9\], at x = 1 and x = 2 are equal.

If for the function 

\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]


Find the derivative of the function f defined by f (x) = mx + c at x = 0.


Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ


Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`


If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.


If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`


If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.


Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`


If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that"  y^2 + "dy"/"dx"` = 0.


Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`


Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)


If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.


If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.


Find the nth derivative of the following:

`(1)/x`


Find the nth derivative of the following : eax+b 


Find the nth derivative of the following:

y = e8x . cos (6x + 7)


Choose the correct option from the given alternatives :

If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?


Choose the correct option from the given alternatives :

If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........


Choose the correct option from the given alternatives :

If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are


Differentiate the following w.r.t. x : `sin^2[cot^-1(sqrt((1 + x)/(1 - x)))]`


Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`


Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`


DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.


Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`


Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0 


Find `"dy"/"dx"` if, yex + xey = 1 


Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`


Solve the following:

If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.


If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`


If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`


If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.


If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______


State whether the following statement is True or False:

If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))`


y = `e^(x3)`


If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.


If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`


If log(x + y) = log(xy) + a then show that, `dy/dx = (−y^2)/x^2`


Find `dy/dx"if", x= e^(3t), y=e^sqrtt`


If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`


If log(x + y) = log(xy) + a, then show that `dy/dx = (-y^2)/x^2`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×