Advertisements
Advertisements
Question
Show that `bb("dy"/"dx" = y/x)` in the following, where a and p are constant:
xpy4 = (x + y)p+4, p ∈ N
Solution
xpy4 = (x + y)p+4
Taking log
log(xpy4) = log(x + y)p+4
logxp + logy4 = (p + 4) log(x + y)
p log x + 4 log y = (p + 4) log(x + y)
Differentiating both sides w.r.t. x, we get
`p."d"/"dx"logx + 4*"d"/"dx"logy = (p + 4)"d"/"dx"log(x + y)`
`p/x + 4(1)/y"dy"/"dx" = (p + 4)(1)/(x + y)(1 + "dy"/"dx")`
`"p"/4 + 4/y"dy"/"dx" = ((p + 4))/((x + y)) + (p + 4)/((x + y))"dy"/"dx"`
`"dy"/"dx"[4/y - ((p + 4))/((x + y))] = (p + 4)/(x + y) - p/x`
`"dy"/"dx"[(4(x + y) -y(p + 4))/(y(x + y))] = (x(p + 4) -p(x + y))/(x(x + y)`
`"dy"/"dx"[(4x + 4y - py - 4y)/(y(x + y))] = (px + 4x - px - py)/(x(x + y)`
`"dy"/"dx"[(4x - py)/y] = (4x - py)/x`
`"dy"/"dx" = y/x`
APPEARS IN
RELATED QUESTIONS
Differentiate the following w.r.t.x:
`(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`
Differentiate the following w.r.t.x: cos(x2 + a2)
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: `e^(3sin^2x - 2cos^2x)`
Differentiate the following w.r.t.x: sec[tan (x4 + 4)]
Differentiate the following w.r.t.x: [log {log(logx)}]2
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(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`
Differentiate the following w.r.t.x:
`(x^2 + 2)^4/(sqrt(x^2 + 5)`
Differentiate the following w.r.t. x : `cot^-1[cot(e^(x^2))]`
Differentiate the following w.r.t.x:
tan–1 (cosec x + cot x)
Differentiate the following w.r.t. x :
`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`
Differentiate the following w.r.t. x : `"cosec"^-1[(10)/(6sin(2^x) - 8cos(2^x))]`
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((8x)/(1 - 15x^2))`
Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`
Differentiate the following w.r.t. x : `tan^-1((a + btanx)/(b - atanx))`
Differentiate the following w.r.t. x :
`tan^-1((5 -x)/(6x^2 - 5x - 3))`
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 : `x^(e^x) + (logx)^(sinx)`
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `cos^-1((7x^4 + 5y^4)/(7x^4 - 5y^4)) = tan^-1a`
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sin((x^3 - y^3)/(x^3 + y^3))` = a3
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 y = sin−1 (2x), find `("d"y)/(""d"x)`
If y = `tan^-1[sqrt((1 + cos x)/(1 - cos x))]`, find `("d"y)/("d"x)`
Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x
If f(x) = 3x - 2 and g(x) = x2, then (fog)(x) = ________.
A particle moves so that x = 2 + 27t - t3. The direction of motion reverses after moving a distance of ______ units.
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 ______
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 ______.
The value of `d/(dx)[tan^-1((a - x)/(1 + ax))]` is ______.
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. x.
