Answer:
f'(x) = x/2
Explanation:
Given the function
f(x) = (1/4)x²
f'(x) will give the first derivative of the function
Using the differentiation formula
Generally, if f(y) = ayⁿ
f'(y) = nay^n-1
Applying this to differentiate the given function,
f'(x) = 2(1/4)x^2-1
f'(x) = (1/2)x
When x = 2
f'(2) = (1/2)×2
f'(2) = 1
When x = 3
f'(3) = (1/2)×3
f'(3) = 3/2
When x = 4
f'(4) = (1/2)×4
f'(4) = 2
Based on the answers, it can be seen that the values keeps increasing arithmetically. The values 1, 1 1/2, and 2 are in arithmetic progression.
The formula for calculating the nth term Tn of an arithmetic progression is expressed as:
Tn = a+(n-1)d
a is the first term of the sequence
n is the number of the terms,
d is the common difference
According to the sequence
a = 1/2, d = 3/2 - 1 = 2 - 3/2 = 1/2
Note that we started with when x=2, the first term will be at when x = 1 which will give 1/2 hence the reason for a = 1/2 instead of 1
Substituting the values in the formula
Tn = 1/2+(n-1)1/2
Tn = 1/2+n/2-1/2
Tn = n/2
Therefore the formula for f'(x) can be expressed as x/2
f'(x) = x/2