Advertisements
Advertisements
प्रश्न
Find the nth derivative of the following : cos x
Advertisements
उत्तर
Let y = cos x
Then `"dy"/"dx" = "d"/"dx"(cosx)`
= `-sinx`
= `cos(pi/2 + x)`
`(d^2y)/(dx^2) = "d"/"dx"(-sinx)`
= `-cosx`
= cos(π + x)
= `cos((2pi)/2 + x)`
`(d^3y)/(dx^3) = "d"/"dx"(-cosx)`
= `-"d"/"dx"(cosx)`
= – ( – sin x)
= sin x
= `cos((3pi)/2 + x)`
In general, the nth order derivative is given by
`(d^ny)/(dx^n) = cos((npi)/2 + x)`.
APPEARS IN
संबंधित प्रश्न
Find `bb(dy/dx)` in the following:
2x + 3y = sin y
Find `bb(dy/dx)` in the following:
sin2 y + cos xy = k
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
Find `bb(dy/dx)` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
Find `"dy"/"dx"` if x = at2, y = 2at.
Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
Differentiate xx w.r.t. xsix.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
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 : apx+q
Find the nth derivative of the following : sin (ax + b)
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 x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1:
| x | f(x) | g(x) | f')x) | g'(x) |
| 0 | 1 | 5 | `(1)/(3)` | |
| 1 | 3 | – 4 | `-(1)/(3)` | `-(8)/(3)` |
(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...
Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
If x = `e^(x/y)`, then show that `dy/dx = (x - y)/(xlogx)`
If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.
If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.
Find `"dy"/"dx"` if, yex + xey = 1
Find `"dy"/"dx"` if, xy = log (xy)
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
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"))`
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
y = `e^(x3)`
Find `dy/dx` if , x = `e^(3t), y = e^(sqrtt)`
Find `dy/dx if , x = e^(3t) , y = e^sqrtt`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Find `dy / dx` if, x = `e^(3t), y = e^sqrt t`
Find `dy/dx` if, x = e3t, y = `e^sqrtt`
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`.
