Calc

1. A formal method of finding a gradient function. y average gradient… f(x+h)-f(x) h x x+h x

2. y f(x+h)-f(x) h x x+h x

3. y f(x+h)-f(x) h x x x+h

4. y f(x+h)-f(x) h x x x+h

5. y f(x+h)-f(x) h x x x+h

6. y f(x+h)-f(x) h x x x+h

7. y f(x+h)-f(x) h x x x+h

8. y f(x+h)-f(x) h x x x+h

9. y x x x+h

10. y x x x+h

11. y instantaneous gradient at x x x:x+h

12. As h→0, the average gradient approaches the actual gradient at x Gradient at x As This leads to the formal definition of the derivative

13. Definition: The function f‘(x) defined by: is called the derivative of f with respect to x – or the gradient function of f(x). Using the definition to find the gradient function (or derivative) of f, is sometimes referred to as “…finding the gradient/derivative using First Principles” Let’s use it to find some other gradient functions….