Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x :
etanx + (logx)tanx
उत्तर
Let y = etanx + (logx)tanx
Put u = (logx)tanx
∴ log u = log (log x)tanx = (tan x).(log log x)
Differentiating both sides w.r.t. x, we get
`1/u."du"/"dx" = "d"/"dx"[(tan x).(log logx)]`
= `(tanx)."d"/"dx"(log log x) + (log log x)."d"/"dx"(tan x)`
= `tanx xx 1/logx."d"/"dx"(logx) + (log log x)(sec^2x)`
= `tanx xx 1/logx xx 1/x + (log log x)(sec^2x)`
∴ `"du"/"dx" = u[tanx/(xlogx) + (loglogx)(sec^2x)]`
= `(logx)^(tanx)[tanx(xlogx) + (log log x)(sec^2x)]`
Now, y = etanx + u
∴ `"dy"/"dx" = "d"/"dx"(e^tanx) + "du"/"dx"`
= `e^(tanx)."d"/"dx"(tanx) + "du"/"dx"`
= `e^(tanx).sec^2x + (logx)^(tanx)[tanx/(xlogx) + (log log x)(sec^2x)]`
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t.x:
`sqrt(x^2 + sqrt(x^2 + 1)`
Differentiate the following w.r.t.x: `5^(sin^3x + 3)`
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x:
tan[cos(sinx)]
Differentiate the following w.r.t.x: `sinsqrt(sinsqrt(x)`
Differentiate the following w.r.t.x: `log[sec (e^(x^2))]`
Differentiate the following w.r.t.x: `log_(e^2) (log x)`
Differentiate the following w.r.t.x:
`(x^3 - 5)^5/(x^3 + 3)^3`
Differentiate the following w.r.t.x: (1 + sin2 x)2 (1 + cos2 x)3
Differentiate the following w.r.t.x:
log (sec 3x+ tan 3x)
Differentiate the following w.r.t.x: `cot(logx/2) - log(cotx/2)`
Differentiate the following w.r.t.x: `log[4^(2x)((x^2 + 5)/(sqrt(2x^3 - 4)))^(3/2)]`
Differentiate the following w.r.t.x:
y = (25)log5(secx) − (16)log4(tanx)
Differentiate the following w.r.t. x : `sin^-1(x^(3/2))`
Differentiate the following w.r.t. x : `cot^-1[cot(e^(x^2))]`
Differentiate the following w.r.t. x : `cos^-1(sqrt((1 + cosx)/2))`
Differentiate the following w.r.t. x :
`cos^-1(sqrt(1 - cos(x^2))/2)`
Differentiate the following w.r.t. x :
`cos^-1((1 - x^2)/(1 + x^2))`
Differentiate the following w.r.t. x : `tan^-1((2x)/(1 - x^2))`
Differentiate the following w.r.t. x : `sin^-1((1 - x^2)/(1 + x^2))`
Differentiate the following w.r.t. x : `cos^-1((e^x - e^(-x))/(e^x + e^(-x)))`
Differentiate the following w.r.t. x:
`tan^-1((2x^(5/2))/(1 - x^5))`
Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 2))`
Differentiate the following w.r.t. x : (sin xx)
Differentiate the following w.r.t. x: xe + xx + ex + ee
Differentiate the following w.r.t. x:
`x^(x^x) + e^(x^x)`
Differentiate the following w.r.t. x : (logx)x – (cos x)cotx
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `tan^-1((3x^2 - 4y^2)/(3x^2 + 4y^2))` = a2
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `e^((x^7 - y^7)/(x^7 + y^7)` = a
If y = sin−1 (2x), find `("d"y)/(""d"x)`
If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`
Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x
Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x
If x = `sqrt("a"^(sin^-1 "t")), "y" = sqrt("a"^(cos^-1 "t")), "then" "dy"/"dx"` = ______
Let f(x) = `(1 - tan x)/(4x - pi), x ne pi/4, x ∈ [0, pi/2]`. If f(x) is continuous in `[0, pi/2]`, then f`(pi/4)` is ______.
If y = cosec x0, then `"dy"/"dx"` = ______.
The volume of a spherical balloon is increasing at the rate of 10 cubic centimetre per minute. The rate of change of the surface of the balloon at the instant when its radius is 4 centimetres, is ______
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
Find `(dy)/(dx)`, if x3 + x3y + xy2 + y3 = 81
