Overview
This lecture explains how to find the derivative of a function at a specific point using the power rule in calculus.
Finding the Derivative
- To find the derivative at a point, first compute the derivative function, denoted as f'(x), with respect to x.
- Use the power rule: multiply the exponent by the coefficient and subtract 1 from the exponent.
- For example, the derivative of ( x^3 ) is ( 3x^2 ), and the derivative of a constant like 5 is 0.
Evaluating the Derivative at a Point
- Substitute ( x = 3 ) into the derivative function to get f'(3).
- Calculate ( 3 \times 3^2 = 27 ).
- Subtract 5 (if needed, based on the function structure) to get the final value, which is 22.
- This value represents the instantaneous rate of change of the function at ( x = 3 ).
Key Terms & Definitions
- Derivative — Measures the rate of change of a function with respect to its variable.
- Power Rule — Shortcut for differentiation: for ( x^n ), the derivative is ( nx^{n-1} ).
- Rate of Change — How a function's output changes as its input changes.
Action Items / Next Steps
- Practice using the power rule to differentiate various polynomial functions.
- Evaluate derivatives at specific points for additional practice.