Advertisements
Advertisements
प्रश्न
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Advertisements
उत्तर
Let y = u. v. w = u. (vw) ....(i)
Differentiating (i) both sides w.r.t. x, we get
(i) `dy/dx = u' .(vw) + u d/dx (vw)`
= u'. (vw) + u [v' w + vw']
= u'. v. w + uv w + uvw'
= `(du)/dx. v. w + u. (dv)/dx . w + u.v. (dw)/dx`
(ii) y = u. v. w
Taking log on both sides, we get
log y = log u + log v + log w ....(ii)
Differentiating (ii) both sides w.r.t. x, we get
`1/y dy/dx = 1/u (du)/dx + 1/v (dv)/dx + 1/w (dw)/dx`
`dy/dx = y (1/u (du)/dx + 1/v (dv)/dx + 1/w (dw)/dx)`
= `uvw (1/u (du)/dx + 1/v (dv)/dx + 1/w (dw)/dx)`
= `vw (du)/dx + uw (dv)/dx + uv (dw)/dx`
= `(du)/dx. v. w + u. (dv)/dx .w + u. v (dw)/dx`
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.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
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
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 `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `dy/dx` if y = xx + 5x
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `(d^2y)/(dx^2)` , if y = log x
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Find the second order derivatives of the following : x3.logx
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
If f(x) = logx (log x) then f'(e) is ______
If log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Derivative of `log_6`x with respect 6x to is ______
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
The derivative of x2x w.r.t. x is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
If xy = yx, then find `dy/dx`
