Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Advertisements
उत्तर
x = t + 2sin (πt), y = 3t – cos (πt)
Differentiating x and y w.r.t. t, we get
`"dx"/"dt" = "d"/"dt"[t + 2sin(pit)]`
= `"d"/"dt"(t) + 2."d"/"dt"[sin(pit)]`
= `1 + 2 xx cos(pit)."d"/"dx"(pit)`
= 1 + 2cos(πt) x π x 1
= 1 + 2π cos (πt)
and
`"dy"/"dt" = "d"/"dt"[3t - cos(pit)]`
= `3 xx 1 - [- sin(pit)]."d"/"dt"(pit)`
= 3 + sin (πt) x π x 1
= 3 + π sin (πt)
∴ `"dy"/"dx" = (("dy"/"dt"))/(("dx"/"dt")`
= `(3 + pi sin(pit))/(1 + 2pi cos(pit)`
∴ `(dy/dx)_("at" t = 1/2)`
= `(3 + sin(pi/2))/(1 + 2picos(pi/2)`
= `(3 + pi xx 1)/(1 + 2pi(0)`
= 3 + π.
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
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:
sin2 y + cos xy = k
If for the function
\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]
Write the derivative of f (x) = |x|3 at x = 0.
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
Differentiate tan-1 (cot 2x) w.r.t.x.
If ex + ey = e(x + y), then show that `dy/dx = -e^(y - x)`.
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
Find `dy/dx` if : x = 2 cos t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Differentiate xx w.r.t. xsix.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : cos x
Choose the correct option from the given alternatives :
If y = sin (2sin–1 x), then dx = ........
Choose the correct option from the given alternatives :
If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "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[2tan^-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))]`
If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log 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.
If x5· y7 = (x + y)12 then show that, `dy/dx = y/x`
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
Choose the correct alternative.
If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
Find `(d^2y)/(dy^2)`, if y = e4x
If `tan ((x + y)/(x - y))` = k, 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`
Solve the following.
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if, x = e3t, y = `e^sqrtt`
Find `dy/dx` if, `x = e^(3t), y = e^sqrtt`
