find the derivative using limit definition f(x) = (x ^ 2 - 4)(x ^ 2 - 5)

Question

find the derivative using limit definition

Answer 1

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).