Dari setiap fungsi f(x) berikut ini, cari dan sederhanakan

dy/dx = 1 / (x + 1)^2

Pembahasan

Untuk setiap fungsi f(x), kita akan mencari (f(x+Δx) - f(x)) / Δx dan kemudian mengambil limitnya saat Δx → 0.

Fungsi: y = x / (x + 1)

Langkah 1: Cari f(x + Δx).
f(x + Δx) = (x + Δx) / ((x + Δx) + 1)
f(x + Δx) = (x + Δx) / (x + Δx + 1)

Langkah 2: Hitung (f(x + Δx) - f(x)).
(f(x + Δx) - f(x)) = [(x + Δx) / (x + Δx + 1)] - [x / (x + 1)]
Samakan penyebutnya:
= [(x + Δx)(x + 1) - x(x + Δx + 1)] / [(x + Δx + 1)(x + 1)]
= [x^2 + x + xΔx + Δx - (x^2 + xΔx + x)] / [(x + Δx + 1)(x + 1)]
= [x^2 + x + xΔx + Δx - x^2 - xΔx - x] / [(x + Δx + 1)(x + 1)]
= Δx / [(x + Δx + 1)(x + 1)]

Langkah 3: Sederhanakan Δy / Δx.
Δy / Δx = [Δx / ((x + Δx + 1)(x + 1))] / Δx
Δy / Δx = 1 / [(x + Δx + 1)(x + 1)]

Langkah 4: Cari dy/dx dengan mengambil limit saat Δx → 0.
dy/dx = lim (Δx→0) [1 / ((x + Δx + 1)(x + 1))]
dy/dx = 1 / ((x + 0 + 1)(x + 1))
dy/dx = 1 / ((x + 1)(x + 1))
dy/dx = 1 / (x + 1)^2

Jadi, (f(x+Δx)-f(x))/Δx = 1 / ((x + Δx + 1)(x + 1)) dan dy/dx = 1 / (x + 1)^2.