Advertisements
Advertisements
Question
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
Advertisements
Solution
x = 2cos4(t + 3), y = 3sin4(t + 3)
∴ `cos^4(t + 3) = x/(2), sin^4(t + 3) = y/(3)`
∴ `cos^2(t + 3) = sqrt((x)/(2)), sin^2(t + 3) = sqrt((y)/(3)`
∵ cos2(t + 3) + sin2(t + 3) = 1
∴ `sqrt((x)/(2)) + sqrt((y)/(3)` = 1
Differentiating x and y w.r.t. t, we get
`(1)/sqrt(2)"d"/"dx"(sqrt(x)) + (1)/sqrt(3)"d"/"dx"(sqrt(y))` = 0
∴ `(1)/sqrt(2) xx (1)/(2sqrt(x)) + (1)/sqrt(3) xx (1)/(2sqrt(y))."dy"/"dx"` = 0
∴ `(1)/(2sqrt(3).sqrt(y))."dy"/"dx" = -(1)/(2sqrt(2).sqrt(x)`
∴ `"dy"/"dx" = -(sqrt(3).sqrt(y))/(sqrt(2).sqrt(x)`
= `-sqrt((3y)/(2x)`.
APPEARS IN
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
`(x + 1/x)^x + x^((1+1/x))`
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
If y = (log x)x + xlog x, find `"dy"/"dx".`
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Find the second order derivatives of the following : x3.logx
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
Derivative of loge2 (logx) with respect to x is _______.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`d/dx(x^{sinx})` = ______
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`2^(cos^(2_x)`
`8^x/x^8`
`log [log(logx^5)]`
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `9^(log_3x)`, find `dy/dx`.
