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
And using the result, find the value of y at :
X_0 = 2
X_0 = 1/2
X_0 = 0
To find the derivative of the function f(x) = 3x^2 + 2x + 1 with respect to x, we take the derivative of each term separately.
f'(x) = d/dx (3x^2) + d/dx (2x) + d/dx (1)
= 6x + 2
Now, to find the value of the derivative at x = x_0, we substitute x_0 into f'(x):
f'(x_0) = 6(x_0) + 2
a) When x_0 = 2:
f'(2) = 6(2) + 2 = 12 + 2 = 14
b) When x_0 = 1/2:
f'(1/2) = 6(1/2) + 2 = 3 + 2 = 5
c) When x_0 = 0:
f'(0) = 6(0) + 2 = 0 + 2 = 2
Therefore, the values of the derivative at x_0 = 2, x_0 = 1/2, and x_0 = 0 are 14, 5, and 2 respectively.