Study Guide for Derivative Calculation of F(x)

Function Definition:
- Given the function is defined as:
- F(x) = 3x f'(x) - f(x)
Need to Find:
- Calculate the derivative F'(2).
- We use the product rule and chain rule to differentiate.

Product Rule Formula:
- The product rule states that for two functions u(x) and v(x):
- (u v)' = u'v + uv'
- For the term 3x f'(x), let:
- u(x) = 3x and v(x) = f'(x)
Derivatives of u and v:
- Calculate u'(x):
- u'(x) = 3
- Calculate v'(x), which is f''(x) by the definition of derivatives:
- v'(x) = f''(x)

Applying the Product Rule:
- The derivative of 3x f'(x) is:
- (3x f'(x))' = u'v + uv' = (3)(f'(x)) + (3x)(f''(x)) = 3f'(x) + 3x f''(x)
- The derivative of -f(x) is:
- (-f(x))' = -f'(x)
- Therefore, for F(x) = 3x f'(x) - f(x):
- F'(x) = (3f'(x) + 3x f''(x)) - f'(x)
- F'(x) = 2f'(x) + 3x f''(x)
Evaluating F'(2):
- Substitute x = 2, f'(2) = -4, and f''(2) = 1 into our derivative equation:
- Calculation steps:
- F'(2) = 2f'(2) + 3(2)f''(2)
- F'(2) = 2(-4) + 6(1)
- F'(2) = -8 + 6
- F'(2) = -2