Advertisements
Advertisements
प्रश्न
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Advertisements
उत्तर
Let, y = `sqrt(((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5)))` ...(1)
or, y = `[((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5))]^(1/2)`
Taking logarithm of both sides,
log y = `1/2 ((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5))` ...[∵ log mn = n log m]
Or log y = `1/2 log (x - 1) (x - 2) - 1/2 log (x - 3) (x - 4) (x - 5) ...[∵ log m/n = log m - log n]`
= `1/2 [log (x- 1) + log (x - 2)] - 1/2 [log (x - 3) + log (x - 4) + log (x - 5)]` ...[∵ log m . n = log m + log n]
Representing both sides by x,
`1/y dy/dx = 1/2 [d/dx log (x - 1) + d/dx log (x - 2)] - 1/2 [d/dx log (x - 3) + d/dx log (x - 4) + d/dx log (x - 5)]`
= `1/2 y [1/(x - 1) d/dx (x - 1) + 1/(x - 2) d/dx (x - 2)] - 1/2 y [1/(x - 3) d/dx (x - 3) + 1/(x - 4) d/dx (x - 4) + 1/(x - 5) d/dx (x - 5)]`
= `1/2 y [1/(x - 1) + 1/(x - 2)] - 1/2 y [1/(x - 3) + 1/(x - 4) + 1/(x - 5)]`
= `1/2 y [1/(x - 1) + 1/(x - 2) - 1/(x - 3) - 1/(x - 4) - 1/(x - 5)]`
Putting the value of y in equation (1),
`dy/dx = 1/2 sqrt(((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5))) [1/(x - 1) + 1/(x - 2) - 1/(x - 3) - 1/(x - 4) - 1/(x - 5)]`
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Differentiate the function with respect to x.
(log x)cos x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
yx = xy
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
- By using the product rule.
- By expanding the product to obtain a single polynomial.
- By logarithmic differentiation.
Do they all give the same answer?
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `(d^2y)/(dx^2)` , if y = log x
Find `"dy"/"dx"` if y = xx + 5x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following: log (ax + b)
Find the nth derivative of the following : log (2x + 3)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Derivative of `log_6`x with respect 6x to is ______
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Find `dy/dx`, if y = (log x)x.
If xy = yx, then find `dy/dx`
