Advertisements
Advertisements
प्रश्न
Find `bb(dy/dx)` for the given function:
xy + yx = 1
Advertisements
उत्तर
xy + yx = 1 ....(i)
Differentiating (i) w.r.t. x we get
`d/dx (x^y) + d/dx (y^x)` = 0 ...(ii)
Let u = xy
Taking log on both sides, we get
log u = y log x ....(iii)
Differentiating the above w.r.t. x, we get
`1/u (du)/dx = d/dx y log x`
= `y d/dx log x +log x d/dx (y)`
= `y * 1/x + log x * dy/dx`
= `y/x + log x dy/dx`
`therefore (du)/dx = u [y/x + log x dy/dx]`
= `x^y [y/x + log x dy/dx]` ...(iv)
Let v = yx
⇒ log v = x log y ... (v)
Differentiating the above w.r.t. x, we get
`1/v (dv)/dx = d/dx x log y`
= `log y d/dx (x) + x d/dx (log y)`
= `log y xx 1 + x xx 1/y dy/dx`
= `log y + x/y dy/dx`
`therefore (dv)/dx = v[log y + x/y dy/dx]`
= `y^x [log y + x/y dy/dx]` .....(vi)
Substituting the values of (iv) and (vi) in (ii), we get
`x^y [y/x + log x dy/dx] + y^x [log y + x/y dy/dx] = 0`
`(x^y log x + xy^(x - 1)) dy/dx = - (y^x log y + yx^(y - 1))`
`dy/dx = -(y^x log y + yx^(y - 1))/(x^y log x + xy^(x - 1))`
APPEARS IN
संबंधित प्रश्न
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
yx = xy
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 cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
Find the second order derivatives of the following : log(logx)
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
Derivative of loge2 (logx) with respect to x is _______.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
Derivative of `log_6`x with respect 6x to is ______
`2^(cos^(2_x)`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` 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 `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
The derivative of x2x w.r.t. x is ______.
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 ______.
For which type of function is logarithmic differentiation especially useful?
After differentiating \[\log y=v(x)\cdot\log[u(x)]\], which equation is obtained?
What is \[\frac{1}{y}\cdot\frac{dy}{dx}\] for \[y=\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
After taking logarithms in logarithmic differentiation, which rules are used to simplify products, quotients and powers before differentiation?
Which derivative correctly represents differentiating \[\ln y\] carefully?
What condition must be ensured for an expression inside logarithm?
