The value of (f(x + h) − f(x))/h is 1 - 6h - 12x.
For any function, say f(x) = x + a, the value of f(x+b) = (x + b) + a or (x + b + a)
Here, we are given that
f(x) = x−6x^2
and we need to find the value of (f(x + h) − f(x))/h
Firstly,
f(x + h) = (x+h) - 6(x+h)^2
Now, f(x + h) − f(x) = (x+h) - 6(x+h)^2 - (x−6x^2)
= (x+h) - 6(x+h)^2 - x + 6x^2
= x - x + h - 6(x^2 + h^2 + 2xh) + 6x^2
= h - 6x^2 - 6h^2 - 12xh + 6x^2
= h - 6x^2 + 6x^2 - 6h^2 - 12xh
= h - 6h^2 - 12xh
= h( 1- 6h - 12x)
Thus, (f(x + h) − f(x))/h = [h( 1- 6h - 12x)] / h
= 1 - 6h - 12x
Therefore, the value of (f(x + h) − f(x))/h is 1 - 6h - 12x.
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