Derivative of (x²-4)(x²-5) from first principles.
Let y=f(x).
We consider a small increment h in x which creates a corresponding increase k in y.
y+k=((x+h)²-4)((x+h)²-5).
y+k=(x²+2xh-4)(x²+2xh-5). h² is ignored as insignificantly small.
y+k=x⁴+4x³h-9x²-18xh+20.
To find k, we subtract f(x):
k=x⁴+4x³h-9x²-18xh+20-(x⁴-9x²+20).
k=4x³h-18xh.
k/h=4x³-18x.
In the limit as h,k→0, h/k=dy/dx=4x³-18x. This is the derivative of f(x).