Find the second derivatives for each of the following functions
a. f(x) =x2
To find the second derivative of f(x) = x^2, we start by finding the first derivative and then take the derivative of that result:
First derivative of f(x) = x^2:
f'(x) = 2x
Second derivative of f(x) = 2x:
f''(x) = 2
Therefore, the second derivative of f(x) = x^2 is f''(x) = 2.
To find the second derivative of the function f(x) = x^2, we need to take the derivative twice.
Step 1: Find the first derivative.
The first derivative of f(x) = x^2 is found by applying the power rule.
f'(x) = 2x.
Step 2: Find the second derivative.
To find the second derivative, we need to take the derivative of the first derivative.
f''(x) = (d/dx)(2x).
Applying the power rule again, where the derivative of x^n is n*x^(n-1), we get:
f''(x) = 2.
Therefore, the second derivative of f(x) = x^2 is f''(x) = 2.