Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Advertisements
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 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:
2x + 3y = sin y
Find `bb(dy/dx)` in the following:
ax + by2 = cos y
Find `bb(dy/dx)` in the following:
sin2 y + cos xy = k
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
If for the function
\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
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 x^3 + y^2 + xy = 7`
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
If ex + ey = e(x + 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 = sinθ, y = tanθ
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
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 `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : `(1)/(3x - 5)`
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(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 x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Solve the following :
f(x) = –x, for – 2 ≤ x < 0
= 2x, for 0 ≤ x < 2
= `(18 - x)/(4)`, for 2 < x ≤ 7
g(x) = 6 – 3x, for 0 ≤ x < 2
= `(2x - 4)/(3)`, for 2 < x ≤ 7
Let u (x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v' (1) and w'(1). If it doesn't exist, then explain why?
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)))]`
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
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, xy = log (xy)
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" = ?`
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
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`
