Jika f(x)=3x^6+5x+6, dengan menggunakan limit, maka

f'(x) = 18x^5 + 5

Pembahasan

Untuk menentukan f'(x) dari f(x) = 3x^6 + 5x + 6 menggunakan definisi limit turunan, kita gunakan rumus:
f'(x) = lim(h->0) [f(x+h) - f(x)] / h

Substitusikan f(x+h) dan f(x):
f(x+h) = 3(x+h)^6 + 5(x+h) + 6
f(x+h) = 3(x^6 + 6x^5h + 15x^4h^2 + ...) + 5x + 5h + 6

[f(x+h) - f(x)] = [3(x^6 + 6x^5h + 15x^4h^2 + ...) + 5x + 5h + 6] - [3x^6 + 5x + 6]
[f(x+h) - f(x)] = 3x^6 + 18x^5h + 45x^4h^2 + ... + 5x + 5h + 6 - 3x^6 - 5x - 6
[f(x+h) - f(x)] = 18x^5h + 45x^4h^2 + ... + 5h

[f(x+h) - f(x)] / h = (18x^5h + 45x^4h^2 + ... + 5h) / h
[f(x+h) - f(x)] / h = 18x^5 + 45x^4h + ... + 5

Sekarang, ambil limit saat h mendekati 0:
f'(x) = lim(h->0) [18x^5 + 45x^4h + ... + 5]
f'(x) = 18x^5 + 0 + ... + 5
f'(x) = 18x^5 + 5