Advertisements
Advertisements
Question
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Solution
Given x7.y9 =(x+y)16
Taking log on both sides
Log(x7.y9) = log(x+y)16
7 log x + 9 log y - 16 log (x+y)
Differentiating w.r.t.x
`7. 1/"x" + 9. 1/"y" "dy"/"dx" = 16 . 1/("x + y") (1 + "dy"/"dx")`
`=> 7/"x" + 9/"y" . "dy"/"dx" = 16/("x + y") + 16/("x + y") "dy"/"dx"`
`=> 9/"y" "dy"/"dx" - 16/("x + y") "dy"/"dx" = 16/("x + y") - 7/"x"`
`=>"dy"/"dx" [9/"y" - 16/"x + y"] = (16 "x" - 7 ("x + y"))/("x" ("x + y"))`
`=>"dy"/"dx" [(9"x" + 9"y" - 16"y")/("y"("x" + "y"))] = (16"x" - 7"x" - 7"y")/"x (x + y)"`
`=> "dy"/"dx" [(9"x" - 7"y")/("y" ("x + y"))] = (9"x" - 7"y")/"x (x +y)"`
`=> "dy"/"dx" = (9"x" - 7"y")/("x (x + y)") xx ("y" ("x + y"))/(9"x" - 7"y") = "y"/"x"`
`=> "dy"/"dx" = "y"/"x"`
APPEARS IN
RELATED QUESTIONS
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`
Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`
Find the second order derivative of the function.
x . cos x
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
ex sin 5x
Find the second order derivative of the function.
e6x cos 3x
Find the second order derivative of the function.
tan–1 x
Find the second order derivative of the function.
log (log x)
If y = cos–1 x, Find `(d^2y)/dx^2` in terms of y alone.
If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0
If y = Aemx + Benx, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`
Find `("d"^2"y")/"dx"^2`, if y = log (x).
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
`sin xy + x/y` = x2 – y
tan–1(x2 + y2) = a
(x2 + y2)2 = xy
If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let,
A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`
and B(x) = [A(x)]T A(x). Then determinant of B(x) ______
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.
If x = a cos t and y = b sin t, then find `(d^2y)/(dx^2)`.
Read the following passage and answer the questions given below:
|
The relation between the height of the plant ('y' in cm) with respect to its exposure to the sunlight is governed by the following equation y = `4x - 1/2 x^2`, where 'x' is the number of days exposed to the sunlight, for x ≤ 3.
|
- Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
- Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, y = `e^((2x + 1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
Find `(d^2y)/dx^2, "if" y = e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x+1))`
