Advertisements
Advertisements
प्रश्न
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
Advertisements
उत्तर
Given, (cos x)y = (cos y)x
Taking logarithm of both sides,
log (cos x)y = log (cos y)x
y log cos x = x log cos y
Differentiating both sides with respect to x,
`y d/dx log cos x + log cos x d/dx (y)= x d/dx log cos y + log cos y d/dx (x)`
⇒ `y * 1/(cos x) d/dx cos x + log cos x * dy/dx= x * 1/(cos y) d/dx cos y + log cos y xx 1`
⇒ `y * 1/(cos x) (- sin x) + log cos x. dy/dx = x 1/(cos y) (-sin y) dy/dx + log cos y`
⇒ `-y tan x + log cos x dy/dx = - x tan y dy/dx + log cos y`
⇒ `log cos x dy/dx + x tan y dy/dx = log cos y + y tan x`
⇒ `dy/dx (log cos x + x tan y) = log cos y + y tan x`
`therefore dy/dx = (log cos y + y tan x)/ (log cos x + x tan y)`
APPEARS IN
संबंधित प्रश्न
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Differentiate the function with respect to x.
(log x)cos x
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))`
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `"dy"/"dx"` if y = xx + 5x
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
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 = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (2x + 3)
If f(x) = logx (log x) then f'(e) is ______
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 x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
`d/dx(x^{sinx})` = ______
`2^(cos^(2_x)`
`8^x/x^8`
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
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`.
The derivative of log x with respect to `1/x` is ______.
If xy = yx, then find `dy/dx`
