Advertisements
Advertisements
प्रश्न
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
Advertisements
उत्तर
log (x + y) = log(xy) + p
∴ log( x + y) = logx + logy + p
Differentiating both sides w.r.t. x, we get
`(1)/(x + y)."d"/"dx"(x + y) = (1)/x + (1)/y."dy"/"dx" + 0`
∴ `(1)/(x + y)(1 + "dy"/"dx") = (1)/x + (1)/y."dy"/"dx"`
∴ `(1)/(x + y) + (1)/(x + y)."dy"/"dx" = (1)/x + (1)/y."dy"/"dx"`
∴ `(1/(x + y) - 1/y)"dy"/"dx" = (1)/x - (1)/(x + y)`
∴ `[(y - x - y)/(y(x + y))]"dy"/"dx" = (x + y - x)/(x(x + y)`
∴ `[(-x)/(y(x + y))]"dy"/"dx" = y/(x(x + y)`
∴ `(-x/y)"dy"/"dx" = y/x`
∴ `"dy"/"dx" = -y^2/x^2`.
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
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.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Find `bb(dy/dx)` for the given function:
yx = xy
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `dy/dx` if y = xx + 5x
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
Find the second order derivatives of the following : log(logx)
If f(x) = logx (log x) then f'(e) is ______
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 y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
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.
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 = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Derivative of `log_6`x with respect 6x to is ______
`2^(cos^(2_x)`
`log [log(logx^5)]`
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
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 ______.
The derivative of x2x w.r.t. x is ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Evaluate:
`int log x dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
