Advertisements
Advertisements
Question
If x5· y7 = (x + y)12 then show that, `dy/dx = y/x`
Advertisements
Solution
x5· y7 = (x + y)12
Taking logarithm of both sides, we get
`log (x^5 * y^7) = log (x + y)^12`
∴ log x5 + log y7 = 12 log (x + y)
∴ 5 log x + 7 log y = 12 log (x + y)
Differentiating both sides w.r.t. x, we get
`5. 1/x + 7. 1/y * dy/dx = 12 * 1/(x + y) * d/dx (x + y)`
∴ `5/x + 7/y * dy/dx = 12/(x + y) [1 + dy/dx]`
∴ `5/x + 7/y * dy/dx = 12/(x + y) + 12/(x + y) * dy/dx`
∴ `[7/y - 12/(x + y)] dy/dx = 12/(x + y) - 5/x`
∴ `[(7x + 7y - 12y)/(y (x + y))] dy/dx = (12x - 5x - 5y)/(x(x + y))`
∴ `[(7x - 5y)/(y(x + y))] dy/dx = [(7x - 5y)/(x(x + y))]`
∴ `dy/dx = [(7x - 5y)/(x(x + y))] xx (y(x + y))/(7x - 5y)`
∴ `dy/dx = y/x`
APPEARS IN
RELATED QUESTIONS
If y=eax ,show that `xdy/dx=ylogy`
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
If f (x) = |x − 2| write whether f' (2) exists or not.
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
Differentiate tan-1 (cot 2x) w.r.t.x.
Discuss extreme values of the function f(x) = x.logx
If ex + ey = e(x + y), then show that `dy/dx = -e^(y - x)`.
Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
DIfferentiate x sin x w.r.t. tan x.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
Find `dy/dx` if, x = `e^(3t)`, y = `e^sqrtt`
If log (x + y) = log (xy) + a then show that, `dy/dx = (−y^2)/x^ 2`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log(x + y) = log(xy) + a then show that, `dy/dx=(-y^2)/x^2`
