Advertisements
Advertisements
Question
Differentiate the following w.r.t. x : `x^(e^x) + (logx)^(sinx)`
Solution
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
RELATED QUESTIONS
Differentiate the following w.r.t.x: `sqrt(tansqrt(x)`
Differentiate the following w.r.t.x: cot3[log(x3)]
Differentiate the following w.r.t.x: log[cos(x3 – 5)]
Differentiate the following w.r.t.x:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x: (1 + 4x)5 (3 + x −x2)8
Differentiate the following w.r.t.x: log[tan3x.sin4x.(x2 + 7)7]
Differentiate the following w.r.t. x : cosec–1 (e–x)
Differentiate the following w.r.t. x : `sin^4[sin^-1(sqrt(x))]`
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 : `tan^-1[(1 - tan(x/2))/(1 + tan(x/2))]`
Differentiate the following w.r.t. x : `tan^-1[(1 + cos(x/3))/(sin(x/3))]`
Differentiate the following w.r.t. x : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x : `"cosec"^-1[(10)/(6sin(2^x) - 8cos(2^x))]`
Differentiate the following w.r.t. x : `tan^-1((2x)/(1 - x^2))`
Differentiate the following w.r.t. x : cos–1(3x – 4x3)
Differentiate the following w.r.t. x : `cos^-1((e^x - e^(-x))/(e^x + e^(-x)))`
Differentiate the following w.r.t. x :
`cos^-1 ((1 - 9^x))/((1 + 9^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((1 + 35x^2)/(2x))`
Differentiate the following w.r.t. x : `tan^-1((2sqrt(x))/(1 + 3x))`
Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 2))`
Differentiate the following w.r.t. x : `(x^5.tan^3 4x)/(sin^2 3x)`
Differentiate the following w.r.t. x : (sin xx)
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Differentiate the following w.r.t. x : `10^(x^(x)) + x^(x(10)) + x^(10x)`
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants: `log((x^20 - y^20)/(x^20 + y^20))` = 20
If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ______.
Differentiate y = etanx w.r. to x
If f(x) is odd and differentiable, then f′(x) is
If x = `sqrt("a"^(sin^-1 "t")), "y" = sqrt("a"^(cos^-1 "t")), "then" "dy"/"dx"` = ______
If `t = v^2/3`, then `(-v/2 (df)/dt)` is equal to, (where f is acceleration) ______
y = {x(x - 3)}2 increases for all values of x lying in the interval.
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 x = eθ, (sin θ – cos θ), y = eθ (sin θ + cos θ) then `dy/dx` at θ = `π/4` is ______.
