1. For a differentiable function f, f' be the funcyion defined by f'(x) lim h->0 =f(x+h)-f(x-h)/(h)
a: Determine f'(x) for f(x)=x
b: Determine f'(x) for f(x)=x^2
I will do the 2nd one, ....
f'(x) = lim ( (x+h)^2 - x^2 )/h , as h --> 0
= lim (x^2 + 2xh + h^2 - x^2)/h
= lim (2xh + h^2)/h , as h -->0
= lim 2x + h , as h --> 0
= 2x
for the 1st one, you should get 1
Thanks ^.^
To determine the derivative, f'(x), of a function f(x), we can use the definition of the derivative provided in the question:
f'(x) = lim h->0 (f(x+h) - f(x-h)) / h
Let's apply this definition to find the derivatives of the given functions:
a) f(x) = x
To find f'(x), substitute f(x) = x into the definition:
f'(x) = lim h->0 (x + h - x - h) / h
= lim h->0 0 / h
= 0
So, the derivative of f(x) = x is f'(x) = 0.
b) f(x) = x^2
Again, substitute f(x) = x^2 into the definition:
f'(x) = lim h->0 ((x + h)^2 - (x - h)^2) / h
= lim h->0 (x^2 + 2xh + h^2 - (x^2 - 2xh + h^2)) / h
= lim h->0 (4xh) / h
= 4x
Therefore, the derivative of f(x) = x^2 is f'(x) = 4x.
To summarize,
a) For f(x) = x, f'(x) = 0.
b) For f(x) = x^2, f'(x) = 4x.