Advertisements
Advertisements
Question
Differentiate the following w.r.t. x : `x^(e^x) + (logx)^(sinx)`
Advertisements
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(x^2 + sqrt(x^2 + 1)`
Differentiate the following w.r.t.x: `sqrt(tansqrt(x)`
Differentiate the following w.r.t.x: log[cos(x3 – 5)]
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:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x:
log (sec 3x+ tan 3x)
Differentiate the following w.r.t.x: `(1 + sinx°)/(1 - sinx°)`
Differentiate the following w.r.t.x:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Differentiate the following w.r.t. x:
`(x^2 + 2)^4/(sqrt(x^2 + 5)`
Differentiate the following w.r.t. x : cot–1(4x)
Differentiate the following w. r. t. x.
cos–1(1 – x2)
Differentiate the following w.r.t. x : `sin^-1(x^(3/2))`
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 - cos(x^2))/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 (cosec x + cot x)
Differentiate the following w.r.t. x :
`cot^-1[(sqrt(1 + sin ((4x)/3)) + sqrt(1 - sin ((4x)/3)))/(sqrt(1 + sin ((4x)/3)) - sqrt(1 - sin ((4x)/3)))]`
Differentiate the following w.r.t. x :
`cos^-1 ((1 - 9^x))/((1 + 9^x)`
Differentiate the following w.r.t. x:
`sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
Differentiate the following w.r.t. x:
`tan^-1((2x^(5/2))/(1 - x^5))`
Differentiate the following w.r.t. x :
`tan^(−1)[(2^(x + 2))/(1 − 3(4^x))]`
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 xx)
Differentiate the following w.r.t. x:
`x^(x^x) + e^(x^x)`
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 : x7.y5 = (x + y)12
Show that `bb("dy"/"dx" = y/x)` in the following, where a and p are constant:
xpy4 = (x + y)p+4, p ∈ N
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: `log((x^20 - y^20)/(x^20 + y^20))` = 20
Solve the following :
The values of f(x), g(x), f'(x) and g'(x) are given in the following table :
| x | f(x) | g(x) | f'(x) | fg'(x) |
| – 1 | 3 | 2 | – 3 | 4 |
| 2 | 2 | – 1 | – 5 | – 4 |
Match the following :
| A Group – Function | B Group – Derivative |
| (A)`"d"/"dx"[f(g(x))]"at" x = -1` | 1. – 16 |
| (B)`"d"/"dx"[g(f(x) - 1)]"at" x = -1` | 2. 20 |
| (C)`"d"/"dx"[f(f(x) - 3)]"at" x = 2` | 3. – 20 |
| (D)`"d"/"dx"[g(g(x))]"at"x = 2` | 5. 12 |
If f(x) is odd and differentiable, then f′(x) is
Differentiate sin2 (sin−1(x2)) w.r. to x
Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x
Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x
If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`
Derivative of (tanx)4 is ______
y = {x(x - 3)}2 increases for all values of x lying in the interval.
A particle moves so that x = 2 + 27t - t3. The direction of motion reverses after moving a distance of ______ units.
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 ______
If x = eθ, (sin θ – cos θ), y = eθ (sin θ + cos θ) then `dy/dx` at θ = `π/4` is ______.
