Advertisements
Advertisements
प्रश्न
If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.
Advertisements
उत्तर
Given: x = `e^(x/y)`
Taking log on both the sides,
log x = `log e^(x/y)`
⇒ log x = `x/y log e`
⇒ log x = `x/y` ...[∵ log e = 1] ...(i)
Differentiating both sides w.r.t. x:
`d/dx log x = d/dx (x/y)`
⇒ `1/x = (y xx 1 - x xx dy/dx)/y^2`
⇒ `y^2 = xy - x^2 xx dy/dx`
⇒ `x^2 xx dy/dx = xy - y^2`
⇒ `dy/dx = (y(x - y))/x^2`
⇒ `dy/d = y/x xx ((x - y)/x)`
⇒ `dy/dx = 1/logx xx ((x - y)/x) ...[∵ log x = x/y "from equation (i)"]`
`dy/dx = (x - y)/(xlogx)`
Hence proved.
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^3)`
Differentiate the following w.r.t. x:
sin (tan–1 e–x)
Differentiate the following w.r.t. x:
log (cos ex)
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
Differentiate the function with respect to x:
cos (a cos x + b sin x), for some constant a and b.
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 `"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`
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" = (varphi "n x")/"x",` then the value of y'' (e) is ____________.
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
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 product rule for logarithms?
What is the domain of the exponential function \[y=b^x\]?
What is the range 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 \[y=\log_b x\]?
The graph of \[y=\log_b x\] is a mirror image of which graph reflected perfectly across \[y=x\]?
Differentiate \[e^{-x}\] with respect to \[x\].
For \[x>0\], differentiate \[\sin(\log x)\] with respect to \[x\].
Which list contains the main log laws?
