Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Advertisements
उत्तर
a cos3θ, y = a sin3θ
Differentiating x and y w.r.t. θ, we get
`"dx"/"dθ" = a"d"/"dθ"(cosθ)^3`
= `a xx 3cos^2θ."d"/"dθ"(cosθ)`
= 3a cos2θ(– sinθ)
= – 3a cos2θ sinθ
and
`"dy"/"dθ" = a"d"/"dθ"(sinθ)^3`
= `a xx 3 sin^2θ."d"/"dθ"(sinθ)`
= 3a sin2θ cosθ
∴`"dy"/"dx" = (("dy"/"dθ"))/(("dx"/"dθ")`
= `(3a sin^2θ cosθ)/(-3a cos^2θ sinθ)`
= – tanθ
∴ `(dx/dy)_("at" θ - pi/3)`
= `-tan pi/(3)`
= `-sqrt(3)`
APPEARS IN
संबंधित प्रश्न
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
xy + y2 = tan x + y
Find `bb(dy/dx)` in the following:
x2 + xy + y2 = 100
if `x^y + y^x = a^b`then Find `dy/dx`
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.
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Differentiate e4x + 5 w.r..t.e3x
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"` if x = at2, y = 2at.
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `tan^-1((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 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 : cos x
Find the nth derivative of the following : y = eax . cos (bx + c)
Find the nth derivative of the following:
y = e8x . cos (6x + 7)
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:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
If x5· y7 = (x + y)12 then show that, `dy/dx = y/x`
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
y = `e^(x3)`
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
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`
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`.
