Find a general formula for F''(x) if F(x) = xf(x) and f and f' are differentiable at x.
To find the general formula for F''(x), we need to differentiate F(x) twice. The stated function F(x) = xf(x) involves the product of two functions, x and f(x). We can use the product rule to differentiate F(x) twice.
Let's start by finding the first derivative of F(x):
F'(x) = (x) * (f(x))' + (f(x)) * (x)'
Since f(x) and x are both differentiable functions, we can differentiate them separately. The derivative of x with respect to x is simply 1; the derivative of f(x) with respect to x is denoted as f'(x).
F'(x) = (x) * f'(x) + f(x)
Now, we need to find the second derivative of F(x) by differentiating F'(x) with respect to x:
F''(x) = [(x) * f'(x) + f(x)]'
To simplify this, we'll use the product rule again:
F''(x) = (x)' * f'(x) + (x) * (f'(x))' + (f(x))'
The derivative of x with respect to x is 1, and we can differentiate f'(x) and f(x) separately:
F''(x) = 1 * f'(x) + x * f''(x) + f'(x)
Simplifying further:
F''(x) = 2f'(x) + xf''(x)
Therefore, the general formula for F''(x) is 2f'(x) + xf''(x).