Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x : `x^(e^x) + (logx)^(sinx)`
Advertisements
उत्तर
Let y = `x^(e^x) + (logx)^(sinx)`
Put u = `x^(e^x) and v = (log x)^(sinx)`
Then y = u + v
∴ `"dy"/"dx" = "du"/"dx" + "dv"/"dx"` ...(1)
Take u = `x^(e^x)`
∴ log u = `logx^(e^x) = e^x.logx`
Differentiating both sides w.r.t. x, we get
`1/"u"."du"/"dx" = "d"/"dx"(e^x log x)`
= `e^x"d"/"dx"(logx) + logx "d"/"dx"(e^x)`
= `e^x.(1)/x + (logx)(e^x)`
∴ `"du"/"dx" = "u" [e^x/x + e^x.log x]`
= `e^x. x^(e^x)[1/x + logx]` ...(2)
Also, v = (log x)sinx
∴ log v = log(log x)sinx = (sin x).(log log x)
Differentiating both sides w.r.t. x, we get
`1/"v"."dv"/"dx" = "d"/"dx"[(sin x).(loglogx)]`
= `(sinx)."d"/"dx"[(log log x) + (log logx)."d"/"dx"(sinx)]`
= `sinx xx 1/logx."d"/"dx"(logx) + (log log x).(cos x)`
∴ `"dv"/"dx" = "v"[sinx/logx xx (1)/x + (cos x)(log log x)]`
= `(logx)^(sinx)[sinx/(xlogx) + (cos x)(log log x)]` ...(3)
From (1), (2) and (3), we get
`"dy"/"dx" - e^x.x^(e^x)[1/x + logx] + (logx)^(sinx) [sinx/(xlogx) + (cosx)(log log x)]`.
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t.x:
`(2x^(3/2) - 3x^(4/3) - 5)^(5/2)`
Differentiate the following w.r.t.x: `(8)/(3root(3)((2x^2 - 7x - 5)^11`
Differentiate the following w.r.t.x: cos(x2 + a2)
Differentiate the following w.r.t.x: `log[tan(x/2)]`
Differentiate the following w.r.t.x: `"cosec"(sqrt(cos x))`
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:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x: `(1 + sinx°)/(1 - sinx°)`
Differentiate the following w.r.t.x: `cot(logx/2) - log(cotx/2)`
Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`
Differentiate the following w.r.t. x : cosec–1 (e–x)
Differentiate the following w.r.t. x : cot–1(4x)
Differentiate the following w.r.t. x : `tan^-1(sqrt(x))`
Differentiate the following w.r.t. x : `sin^4[sin^-1(sqrt(x))]`
Differentiate the following w.r.t. x : `cos^-1(sqrt((1 + cosx)/2))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`
Differentiate the following w.r.t. x :
`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`
Differentiate the following w.r.t. x : `tan^-1((2x)/(1 - x^2))`
Differentiate the following w.r.t. x:
`tan^-1((2x^(5/2))/(1 - x^5))`
Differentiate the following w.r.t. x : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
Differentiate the following w.r.t.x:
`cot^-1((1 + 35x^2)/(2x))`
Differentiate the following w.r.t. x :
`tan^-1((5 -x)/(6x^2 - 5x - 3))`
Differentiate the following w.r.t. x :
`(x + 1)^2/((x + 2)^3(x + 3)^4`
Differentiate the following w.r.t. x : `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`
Differentiate the following w.r.t. x : (sin x)x
Differentiate the following w.r.t. x:
`x^(x^x) + e^(x^x)`
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
Differentiate y = `sqrt(x^2 + 5)` w.r. to x
If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`
If y = `tan^-1[sqrt((1 + cos x)/(1 - cos x))]`, find `("d"y)/("d"x)`
Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x
Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x
If f(x) = 3x - 2 and g(x) = x2, then (fog)(x) = ________.
If the function f(x) = `(log (1 + "ax") - log (1 - "bx))/x, x ≠ 0` is continuous at x = 0 then, f(0) = _____.
Derivative of (tanx)4 is ______
If y = `(3x^2 - 4x + 7.5)^4, "then" dy/dx` is ______
The weight W of a certain stock of fish is given by W = nw, where n is the size of stock and w is the average weight of a fish. If n and w change with time t as n = 2t2 + 3 and w = t2 - t + 2, then the rate of change of W with respect to t at t = 1 is ______
If x2 + y2 - 2axy = 0, then `dy/dx` equals ______
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)`
Let f(x) be a polynomial function of the second degree. If f(1) = f(–1) and a1, a2, a3 are in AP, then f’(a1), f’(a2), f’(a3) are in ______.
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. x.
