Miten erotat f (x) = x ^ 3sqrt (x-2) sinx tuotesäännön avulla?

Vastaus:

#f '(x) = 3x ^ 2sqrt (x-2) sinx + (x ^ 3sinx) / (2sqrt (x-2)) + x ^ 3sqrt (x-2) cosx #

Selitys:

Jos #f (x) = g (x) h (x) j (x) #sitten #f '(x) = g' (x) h (x) j (x) + g (x) h '(x) j (x) + g (x) h (x) j (x) #

#G (x) = x ^ 3 #

#G '(x) = 3x ^ 2 #

#h (x) = sqrt (x-2) = (x-2) ^ (1/2) #

#h '(x) = 1/2 * (x-2) ^ (- 1/2) * d / dx x-2 #

#COLOR (valkoinen) (h '(x)) = (x-2) ^ (- 1/2) / 2 * 1 #

#COLOR (valkoinen) (h '(x)) = (x-2) ^ (- 1/2) / 2 #

#COLOR (valkoinen) (h '(x)) = 1 / (2sqrt (x-2)) #

#J (x) = sinx #

#J '(x) = cosx #

#f '(x) = 3x ^ 2sqrt (x-2) sinx + x ^ 3 1 / (2sqrt (x-2)) sinx + x ^ 3sqrt (x-2) cosx #

#f '(x) = 3x ^ 2sqrt (x-2) sinx + (x ^ 3sinx) / (2sqrt (x-2)) + x ^ 3sqrt (x-2) cosx #

Miten erotat f (x) = 2x ^ 2 * e ^ x * sinx tuotesäännön avulla?

2xe ^ x (2sinx + xsinx + xcosx) f '(x) = (2x ^ 2e ^ xsinx)' = (2x ^ 2) 'e ^ xsinx + 2x ^ 2 (e ^ x)' sinx + 2x ^ 2e ^ x (sinx) '= 4xe ^ xsinx + 2x ^ 2e ^ xsinx + 2x ^ 2e ^ xcosx = 2xe ^ x (2sinx + xsinx + xcosx)

Miten erotat f (x) = 2x (x ^ 2-1) tuotesäännön avulla?

2 (3x ^ 2-1) f (x) = 2x (x ^ 2-1) df / dx = 2 (dx / dx. (X ^ 2-1) + xd / dx (x ^ 2-1)) Tuotesääntö: d / dx (uv) = (du / dx) v + u (dv / dx) df / dx = 2 ((x ^ 2-1) + x.2x) df / dx = 2 (x ^ 2 -1 + 2x ^ 2) = 2 (3 x ^ 2-1)

Miten erotat g (x) = (x ^ 2 + 1) (x ^ 2-3x) tuotesäännön avulla?

G '(x) = 4x ^ 3-6x ^ 2 + 2x-2 g (x) = (x ^ 2 + 1) (x ^ 2-2x) Tuotesääntö: d / dx (uv) = (du) / dxv + u (dv) / dx u = (x ^ 2 + 1) du / dx = 2x v = x ^ 2-2x dv / dx = 2x = 2 d / dx (x ^ 2 + 1) (x ^ 2 -2x) = (du) / dxv + u (du) / dx = 2x (x ^ 2-2x) + (x ^ 2 + 1) (2x-2) = 2x ^ 3-4x ^ 2 + 2x ^ 3 -2x ^ 2 + 2x-2 = 4x ^ 3-6x ^ 2 + 2x-2