Advertisements
Advertisements
Question
Differentiate the following:
y = `"e"^(3x)/(1 + "e"^x`
Advertisements
Solution
y = `"e"^(3x)/(1 + "e"^x`
`("d"y)/("d"x) = ((1 + "e"^x) "e"^(3x) (3) - "e"^(3x) (0 + "e"^x))/(1 + "e"^x)^2`
`("d"y)/("d"x) = (3(1 + "e"^x) "e"^(3x) - "e"^(3x) * "e"^x)/(1 + "e"^x)^2`
`("d"y)/("d"x) = (3"e"^(3x) + 3"e"^x "e"^(3x) - "e"^x "e"^(3x))/(1 + "e"^x)^2`
= `(3"e"^(3x) + 2"e"^x "e"^(3x))/(1 + "e"^x)^2`
= `(3"e"^(3x) + 2"e"^(x + 3x))/(1 + "e"^x)^2`
`("d"y)/("d"x) = (3"e"^(3x) + 2"e"^(4x))/(1 + "e"^x)^2`
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
f(x) = x – 3 sin x
Find the derivatives of the following functions with respect to corresponding independent variables:
g(t) = t3 cos t
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `x/(sin x + cosx)`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = (x2 + 5) log(1 + x) e–3x
Differentiate the following:
y = tan 3x
Differentiate the following:
y = cos (tan x)
Differentiate the following:
y = sin (ex)
Differentiate the following:
y = e–mx
Differentiate the following:
y = `"e"^(xcosx)`
Find the derivatives of the following:
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
Find the derivatives of the following:
If cos(xy) = x, show that `(-(1 + ysin(xy)))/(xsiny)`
Find the derivatives of the following:
`tan^-1 = ((6x)/(1 - 9x^2))`
Find the derivatives of the following:
`cos[2tan^-1 sqrt((1 - x)/(1 + x))]`
Find the derivatives of the following:
sin-1 (3x – 4x3)
Find the derivatives of the following:
If sin y = x sin(a + y), the prove that `("d"y)/("d"x) = (sin^2("a" + y))/sin"a"`, a ≠ nπ
Find the derivatives of the following:
If y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y2 when x = 0
Choose the correct alternative:
`"d"/("d"x) (2/pi sin x^circ)` is
Choose the correct alternative:
If y = `1/4 u^4`, u = `2/3 x^3 + 5`, then `("d"y)/("d"x)` is
Choose the correct alternative:
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)` then `("d"y)/("d"x)` is
Choose the correct alternative:
If f(x) = `{{:("a"x^2 - "b"",", - 1 < x < 1),(1/|x|",", "elsewhere"):}` is differentiable at x = 1, then
