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 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:
`sqrt(e^(sqrtx))`, x > 0
Differentiate the following w.r.t. x:
log (log x), x > 1
Differentiate the following w.r.t. x:
cos (log x + ex), x > 0
Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.
If `"y" ="x"^"x" , "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")`.
The derivative of log10x w.r.t. x is ______.
If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.
Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`
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 `"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"`
If `"y = tan"^-1 [("sin x + cos x")/("cos x - sin x")], "then" "dy"/"dx"` is equal to ____________.
If f(x) = `"log"_("x"^2) ("log x")`, then f(e) is ____________.
Which function is an exponential function?
For positive values of \[x\], which expression grows faster than \[x^n\] for any fixed positive integer \[n\] when \[x\] is sufficiently large?
What is \[\frac{d}{dx}(e^x)\]?
What is \[\frac{d}{dx}(\log x)\] for \[x>0\]?
Which expression gives the power rule for logarithms?
Which inverse property is valid only for \[x>0\]?
What is the domain of the exponential function \[y=b^x\]?
Through which point does every graph of \[y=b^x\] pass?
What is the domain of the logarithmic function \[y=\log_b x\]?
Through which point does every graph of \[y=\log_b x\] pass?
Which pair of graphs are reflections of each other about \[y=x\]?
Differentiate \[\cos^{-1}(e^x)\] with respect to \[x\].
