By first principle, Find the derivative of the function
iii.) f(x) = 3x^2 + 2x + 1
with respect to x at x = x subscript 0
To find the derivative of f(x), we need to use the power rule for derivatives. The power rule states that the derivative of x^n with respect to x is nx^(n-1).
Given f(x) = 3x^2 + 2x + 1, we can find the derivative by applying the power rule to each term:
f'(x) = d/dx [3x^2] + d/dx [2x] + d/dx [1]
f'(x) = 2(3)x^(2-1) + 1(2)x^(1-1) + 0
f'(x) = 6x + 2
Now, to find the derivative at x = x_0, we substitute x_0 into the derivative:
f'(x_0) = 6x_0 + 2
Therefore, the derivative of f(x) with respect to x at x = x_0 is 6x_0 + 2.
In other words
6 x + 2