Advertisements
Advertisements
Question
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
Advertisements
Solution
`(sin "x")^"y" = "x" + "y"`
Take log on both the sides,
`log(sin "x")^"y" = log("x" + "y")`
⇒ `"y" log (sin "x") = log ("x" + "y")` ......(i)
Differentiate (i) w.r.t.x
`log (sin "x")· (d"y")/(d"x") + "y"· (d)/(d"x") [ log(sin "x")] = (d)/(d"x") [log ("x"+"y") ]`
⇒ `log (sin "x")· (d"y")/(d"x") + "y"· (cos "x")/(sin"x") = (1)/(("x"+"y"))· (1+ (d"y")/(d"x"))`
⇒ `(d"y")/(d"x") [ log( sin "x") - (1)/(("x"+"y"))] = (1)/(("x"+"y")) - "y"·cot "x" `
⇒ `(d"y")/(d"x") = (1 - ("xy" + "y"^2)·cot "x")/(("x"+"y")·log (sin "x") -1)`
APPEARS IN
RELATED QUESTIONS
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
xy + yx = 1
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.
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Find `"dy"/"dx"` if y = xx + 5x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
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
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.logx
Find the second order derivatives of the following : log(logx)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
Derivative of loge2 (logx) with respect to x is _______.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
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^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
If y = `9^(log_3x)`, find `dy/dx`.
The derivative of log x with respect to `1/x` is ______.
Find `dy/dx`, if y = (log x)x.
If xy = yx, then find `dy/dx`
