Advertisements
Advertisements
Question
Find `dy/dx` if : x = 2 cos t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`
Advertisements
Solution
2 cos t + cos 2t, y = 2 sin t – sin 2t
Differentiating x and y w.r.t. t, we get
`"dx"/"dt" = "d"/"dt"(2cost + cos2t)`
= `2"d"/"dt"(cost) + "d"/"dt"(cos2t)`
= `2(-sin t) + (- sin 2t)."d"/"dt"(2t)`
= – 2 sin t – sin 2t x 2 x 1
= – 2 sin t – 2 sin 2t and,
`"dy"/"dt" = "d"/"dt"(2sint - sin2t)`
= `2"d"/"dt"(sint) - "d"/"dt"(sin2t)`
= `2cost - cos2t."d"/"dt"(2t)`
= 2 cos t – cos 2t x 2 x 1
= 2 cos t – 2 cos 2t
∴ `"dy"/"dx" = (("dy"/"dt"))/(("dx"/"dt")`
= `(2cost - 2cos2t)/(-2sint - 2sin2t)`
= `(cost - cos2t)/(-sint - sin2t)`
∴ `(dy/dx)_("at" t = pi/4)`
= `(cos pi/4 - cos pi/2)/(-sin pi/4 - sin pi/2)`
= `(1/sqrt(2) - 0)/(-1/sqrt(2) - 1`
= `(-1)/(1 + sqrt(2)`
= `(-1)/(1 + sqrt(2)) xx (1 - sqrt(2))/(1 - sqrt(2)`
= `(-(1 - sqrt(2)))/(1 - 2)`
= `1 - sqrt(2)`.
APPEARS IN
RELATED QUESTIONS
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
x2 + xy + y2 = 100
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
If for the function
\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
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|?
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 \[\lim_{x \to c} f\left( x \right)\]
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Differentiate e4x + 5 w.r..t.e3x
Differentiate tan-1 (cot 2x) w.r.t.x.
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
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`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
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 : eax+b
Find the nth derivative of the following : apx+q
Find the nth derivative of the following : sin (ax + b)
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 f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
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 : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
If log (x + y) = log (xy) + a then show that, `"dy"/"dx" = (- "y"^2)/"x"^2`.
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
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)` = ______
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 `xy + y^2 = tan x + y` is
Find `(d^2y)/(dy^2)`, if y = e4x
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 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 = 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`
