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 function with respect to x.
(log x)cos x
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Differentiate 3x w.r.t. logx3.
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
Derivative of loge2 (logx) with respect to x is _______.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`log [log(logx^5)]`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` 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 y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `9^(log_3x)`, find `dy/dx`.
Find the derivative of `y = log x + 1/x` with respect to x.
