Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
उत्तर
`x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`
Put t = cosθ.
Thenθ = cos–1t.
∴ x = cos–1(4cos3θ – 3cosθ ),
y = `tan^-1(sqrt(1 - cos^2θ)/cosθ)`
∴ x = `cos^-1(cos3θ),y = tan^-1((sinθ)/(cosθ)) = tan^-1(tanθ)`
∴ x = 3θ and y = θ
∴ x = 3cos–1t and y = cos–1t
Differentiating x and y w.r.t. x, we get
`"dx"/"dt" = 3"d"/"dt"(cos^-1)`
= `3 xx (-1)/sqrt(1 - t^2)`
= `(-3)/sqrt(1 - t^2)`
and
`"dy"/"dt" = "d"/"dt"(cos^-1t)`
= `(-1)/sqrt(1 - t^2)`
∴ `"dy"/"dx" = (("dy"/"dt"))/(("dx"/"dt")`
= `(((-1)/(sqrt(1 - t^2))))/(((-3)/(sqrt(1 - t^2)))`
= `(1)/(3)`.
Alternative Method :
x = cos–1 (4t3 – 3t), t = `tan^-1(sqrt(1 - t^2)/t)`
Put t = cosθ.
Then x = cos–1(4cos3θ – 3cosθ),
y = `tan^-1(sqrt(1- cos^2θ)/cosθ)`
∴ x = cos–1 (cos3θ), y = `tan^-1((sinθ)/(cosθ)) = tan^-1(tanθ)`
∴ x = 3θ, y = θ
∴ x = 3y
∴ y = `(1)/(3)x`
∴ `"dy"/"dx" = (1)/(3)"d"/"dx"(x)`
= `(1)/(3) xx 1`
= `(1)/(3)`.
APPEARS IN
संबंधित प्रश्न
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `dy/dx` in the following:
ax + by2 = cos y
Find `dy/dx` in the following.
x3 + x2y + xy2 + y3 = 81
Find `dy/dx` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
If f (x) = |x − 2| write whether f' (2) exists or not.
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
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 x = tan-1t and y = t3 , find `(dy)/(dx)`.
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ
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)`
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
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))`.
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `π/4`.
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.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 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 : eax+b
Find the nth derivative of the following : sin (ax + b)
Find the nth derivative of the following : `(1)/(3x - 5)`
Find the nth derivative of the following:
y = e8x . cos (6x + 7)
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 `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 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:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.
Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0
Find `"dy"/"dx"` if, xy = log (xy)
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` 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 x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
`(dy)/(dx)` of `2x + 3y = sin x` is:-
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.
If 2x + 2y = 2x+y, 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`
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)`
If log(x + y) = log(xy) + a, then show that `dy/dx = (-y^2)/x^2`
