Advertisements
Advertisements
Question
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Solution
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
RELATED QUESTIONS
Find `dy/dx` in the following:
sin2 y + cos xy = k
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
Is |sin x| differentiable? What about cos |x|?
Find `dy/dx if x^3 + y^2 + xy = 7`
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Differentiate e4x + 5 w.r..t.e3x
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
Discuss extreme values of the function f(x) = x.logx
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that" y^2 + "dy"/"dx"` = 0.
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Differentiate xx w.r.t. xsix.
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
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 `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 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 y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
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 `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, 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)` = .........
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
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 : `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(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If `x = e^(x/y)`, then show that `"dy"/"dx" = (x - y)/(xlogx)`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
Find `"dy"/"dx"` if, yex + xey = 1
Solve the following:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
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)` = ______.
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"))`
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
Find `(d^2y)/(dy^2)`, if y = e4x
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`
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 = 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`
Find `dy/dx"if", x= e^(3t), y=e^sqrtt`
Find `dy/(dx) "if" , x = e^(3t), y = e^sqrtt`.
