Advertisements
Advertisements
प्रश्न
If xy = ex–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`
Advertisements
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^x/sinx`
Differentiate the following w.r.t. x:
`e^x + e^(x^2) + "..." + e^(x^5)`
Differentiate the following w.r.t. x:
`sqrt(e^(sqrtx))`, x > 0
Differentiate the following w.r.t. x:
cos (log x + ex), x > 0
Differentiate the function with respect to x:
(log x)log x, x > 1
If `"y" ="x"^"x" , "find" "dy"/"dx"`.
If xy - yx = ab, find `(dy)/(dx)`.
The derivative of log10x w.r.t. x is ______.
If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.
If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`
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" = (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 `"y"^2 = "ax"^2 + "bx + c", "then" "d"/"dx" ("y"^3 "y"_"z") =` ____________.
If `sqrt(("x + y")) + sqrt (("y - x")) = "a", "then" "dy"/"dx" =` ____________.
If `"xy"^2 = "ax"^2 + "bxy" + "y"^2, "then find" "dy"/"dx"`
Which function is an exponential function?
Which expression gives the quotient rule for logarithms?
Which expression gives the power rule for logarithms?
What is the domain of the exponential function \[y=b^x\]?
What is the domain of the logarithmic function \[y=\log_b x\]?
The graph of \[y=\log_b x\] is a mirror image of which graph reflected perfectly across \[y=x\]?
If \[y=b^x\], which equivalent logarithmic statement is correct?
Differentiate \[e^{-x}\] with respect to \[x\].
For \[x>0\], differentiate \[\sin(\log x)\] with respect to \[x\].
Differentiate \[\cos^{-1}(e^x)\] with respect to \[x\].
Which list contains the main log laws?
