Find dydx if x = e3t, y=et. - Mathematics and Statistics

Advertisements
Advertisements
Sum

Find `"dy"/"dx"` if x = `"e"^"3t",  "y" = "e"^(sqrt"t")`.

Advertisements

Solution

x = `"e"^"3t"`

Differentiating both sides w.r.t. t, we get

`"dx"/"dt" = "e"^"3t" * "d"/"dx" ("3t")`

`= "e"^"3t" * (3)`

∴ `"dx"/"dt" = 3"e"^"3t"`

y = `"e"^(sqrt"t")`

Differentiating both sides w.r.t. t, we get

`"dy"/"dt" = "e"^(sqrt"t") * "d"/"dx" (sqrt"t")`

`"dy"/"dt" = "e"^(sqrt"t") * 1/(2 sqrt"t")`

∴ `"dy"/"dx" = ("dy"/"dt")/("dx"/"dt") = "e"^(sqrt"t")/((2 sqrt"t")/(3"e"^"3t"))`

∴ `"dy"/"dx" = 1/(6 sqrt"t")  "e"^(sqrt"t" - "3t")`

  Is there an error in this question or solution?
Chapter 3: Differentiation - Miscellaneous Exercise 3 [Page 100]

APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 3 Differentiation
Miscellaneous Exercise 3 | Q 4.16 | Page 100

RELATED QUESTIONS

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  `dy/dx`

2x + 3y = sin x


Find `dy/dx`

ax + by2 = cos y


Find `dy/dx`

xy + y2 = tan x + y


Find `dx/dy`

x2 + xy + y2 = 100


Find `dy/dx`

x3 + x2y + xy2 + y3 = 81


Find `dy/dx`

sin2 y + cos xy = Π


Find `dy/dx`

sin2 x + cos2 y = 1


if `x^y + y^x = a^b`then Find `dy/dx`


Show that the derivative of the function f given by 

\[f\left( x \right) = 2 x^3 - 9 x^2 + 12x + 9\], at x = 1 and x = 2 are equal.

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).


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.


Write the derivative of f (x) = |x|3 at x = 0.


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 = sin^-1(sqrt(1-x^2))`


Differentiate tan-1 (cot 2x) w.r.t.x.


If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`


If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.


Find `"dy"/"dx"` if x = at2, y = 2at


Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ


Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`


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 = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.


Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`


Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.


Find `dy/dx` if : x = 2 cos t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`


Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`


If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that"  y^2 + "dy"/"dx"` = 0.


Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`


Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.


Differentiate xx w.r.t. xsix.


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 y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.


If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.


If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.


If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 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 : (ax + b)m 


Find the nth derivative of the following : `(1)/x`


Find the nth derivative of the following : apx+q 


Find the nth derivative of the following : cos x


Find the nth derivative of the following : sin (ax + b)


Find the nth derivative of the following : cos (3 – 2x)


Find the nth derivative of the following : `(1)/(3x - 5)`


Find the nth derivative of the following : y = eax . cos (bx + c)


Choose the correct option from the given alternatives : 

Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is 


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 = ........


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)))]`


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((sqrt(x)(3 - x))/(1 - 3x))`


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))`


Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`


If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.


If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.


If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.


DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 + x^2))/(1 - 2x^2)))`.


Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).


Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`


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 y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.


Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0 


Find `"dy"/"dx"` if, yex + xey = 1 


Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`


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 ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?` 


Choose the correct alternative.

If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?


If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`


If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.


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 `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______


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 `2x + 3y = sin x` is:-


`(dy)/(dx)` of `xy + y^2 = tan x + y` is


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`


y = `e^(x3)`


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 ______.


Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to ______.


If `tan ((x + y)/(x - y))` = k, 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`


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`


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 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`


Share
Notifications



      Forgot password?
Use app×