Advertisements
Advertisements
प्रश्न
If xy = ex–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`
उत्तर
We have xy = ex–y
Taking logarithm on both sides, we get
y log x = x – y
⇒ y(1 + log x) = x
i.e. y = `x/(1 + log x)`
Differentiating both sides w.r.t. x, we get\
`("d"y)/("d"x) = ((1 + logx) * 1 - x(1/x))/(1 + logx)^2`
= `logx/(1 + log x)^2`.
APPEARS IN
संबंधित प्रश्न
Differentiate 3x w.r.t. log3x
Differentiate the following w.r.t. x:
`e^(sin^(-1) x)`
Differentiate the following w.r.t. x:
`e^(x^3)`
Differentiate the following w.r.t. x:
sin (tan–1 e–x)
Differentiate the following w.r.t. x:
`log(cos e^x)`
Differentiate the following w.r.t. x:
`e^x + e^(x^2) +... + e^(x^3)`
Differentiate the following w.r.t. x:
log (log x), x > 1
Differentiate the following w.r.t. x:
`cos x/log x, x >0`
Differentiate the following w.r.t. x:
cos (log x + ex), x > 0
Differentiate w.r.t. x the function:
(log x)log x, x > 1
Differentiate w.r.t. x the function:
cos (a cos x + b sin x), for some constant a and b.
If `"y" ="x"^"x" , "find" "dy"/"dx"`.
If xy - yx = ab, find `(dy)/(dx)`.
If `"x" = "e"^(cos2"t") "and" "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.
If yx = ey – x, prove that `"dy"/"dx" = (1 + log y)^2/logy`
Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`
If `"y" = ("x" + sqrt(1 + "x"^2))^"n", "then" (1 + "x"^2) ("d"^2 "y")/"dx"^2 + "x" ("dy")/("dx")` is ____________.
If `"y = a"^"x", "b"^(2"x" -1), "then" ("d"^2"y")/"dx"^2` is ____________.
If `"y" = (varphi "n x")/"x",` then the value of y'' (e) is ____________.
If `"x" = "a" ("cos" theta + theta "sin" theta), "y = a" ("sin" theta - theta "cos" theta), "then" ("d"^2 "y")/("dx"^2) =` ____________.
If `sqrt(("x + y")) + sqrt (("y - x")) = "a", "then" "dy"/"dx" =` ____________.
If `"xy"^2 = "ax"^2 + "bxy" + "y"^2, "then find" "dy"/"dx"`
If `"y = tan"^-1 [("sin x + cos x")/("cos x - sin x")], "then" "dy"/"dx"` is equal to ____________.
The domain of the function defined by f(x) = logx 10 is
