What is the derivative of y = (1 - x)(2 - x)....(n - x) at x = 1?

Solution:

To find the derivative of y = (1-x)(2-x)....(n-x) at x=1 we will differentiate y with respect to 'x'.

We need to find the derivative of y = (1 - x)(2 - x)…….(n - x)

∵ d/dx(uvw) = u’vw + uv’w + uvw’

dy/dx = -1(2 - x)(3 - x)...(n - x) + (1 - x)(-1)(3 - x)...(n - x) +....+ (n - 1 - x)(-1)

All terms that include (1 - x) will become zero at x = 1, and only one term will be left.

⇒ dy/dx = -1(2 - 1)(3 - 1)….(n - 1) + 0 +…., at x = 1

⇒ dy/dx = -1 × 2 × 3 ×...× (n - 1) 

⇒ -1(n - 1)!

Thus, the derivative of y = (1 - x)(2 - x)....(n - x) at x = 1 is equal to (-1)(n - 1)!

What is the derivative of y = (1 - x)(2 - x)....(n - x) at x = 1?

Summary:

The value of y = (1 - x)(2 - x)....(n - x) at x = 1 is - 1(n - 1)!