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Limits to Derivatives Remember the last class? Good! I believe I mentioned that it was a rather important class. Let’s review: There are two ways to use limits to find the slope of a tangent. Method 1: mtangent lim xa f ( x) f (a ) xa Method 2: mtangent lim h 0 f ( a h) f ( a ) h This is the whole basis to calculus! We can use this equation to find the slope for any equation. Example: Find the slope of the curve f(x)=x2-7x+1 at the point (-2,19). a 2, f (a ) 19 f ( a h) f ( a ) h 0 h f (h 2) 19 lim h 0 h (h 2)2 7(h 2) 1 19 lim h 0 h 2 h 4h 4 7 h 14 1 19 lim h 0 h 2 h 11h lim h 0 h h(h 11) lim h 0 h lim(h 11) mtangent lim mtangent mtangent mtangent mtangent mtangent mtangent h 0 mtangent 11 Easy….right? Example: Find the slope of the curve f(x)=x2 at the point (-3,9). a 3, f (a ) 9 f ( a h) f ( a ) h 0 h f (h 3) 9 lim h 0 h (h 3) 2 9 lim h 0 h 2 h 6h 9 9 lim h 0 h 2 h 6h lim h 0 h h(h 6) lim h 0 h lim(h 6) mtangent lim mtangent mtangent mtangent mtangent mtangent mtangent h 0 mtangent 6 Now we can generalize this for any point. Find the slope of the curve f(x)=x2 at the point (x,y). a x, f ( a ) y x 2 f ( a h) f ( a ) h 0 h f (h x) y lim h 0 h (h x)2 x 2 lim h 0 h 2 h 2 xh x 2 x 2 lim h 0 h 2 h 2 xh lim h 0 h h( h 2 x ) lim h 0 h lim(h 2 x) mtangent lim mtangent mtangent mtangent mtangent mtangent mtangent h 0 mtangent 2 x Finding the general rule to describe the functions at any point is called finding the derivative. The derivative is a function that represents the slope of a function at any point. We use two main ways to represent a function. The derivative of a function f(x) is f’(x) [we say: “f prime of x” or simply “the derivative of f”]. df (x) dy Another way to represent the derivative of the relation f (x) y is or [we dx dx say: “the derivative of f”]. Therefore in the problem above we can say: The derivative of f(x)=x2 is f’(x)=2x or dy 2x . dx We now have a general solution for the derivative of f(x)=x2. If we were to do this for other simple polynomials, we may get a pattern. Take a look at this pattern f ( x) c f '( x) 0 f ( x) x f '( x) 1 f ( x) x 2 f '( x) 2 x f ( x) x3 f '( x) 3x 2 f ( x) x 4 f '( x) 4 x3 f ( x) x5 f '( x) 5 x 4 hmmmm…..there’s a pattern it seems. (i) f ( x) x n f '( x) nx n1 lim cf ( x) c lim f ( x) Now if you look back at the rules on limits: x a x a lim f ( x) g ( x) lim f ( x) lim g ( x) x a We can apply them to develop a few more general rules. (ii) f ( x) ax n f '( x) anx n1 Furthermore, (iii) f ( x) ax n bx m f '( x) anx n1 bmx m1 These are some of the most useful rules in calculus this year. You should also remember that (iv) f ( x) c f '( x) 0 This is true since f ( x) c cx0 f '( x) c 0 x01 0 x a x a Let’s use these rules: Example 1:(a) Find the derivative of f ( x) 5 x3 4 x 2 7 x 5 Solution: f '( x) 15x 2 8x 7 (b) Find the slope of the tangent of f ( x ) at the point (2,33). Solution: f '(2) 15(2)2 8(2) 7 51 So the slope when x=2 of the function is 51. Easy Example 2:(a) Find the derivative of f ( x) x4 3x2 8x 3 Solution: f '( x) 4 x3 6 x 8 (b) Find the slope of the tangent of f ( x ) when x=1. Solution: f '(1) 4(1)3 6(1) 8 6 So the slope when x=1 of the function is -6. Easy (again) Notice that this rule would work for any function that can be written as an exponent. Example 3:(a) Find the derivative of f ( x) x 1 Solution: f ( x) x 2 1 12 1 1 12 f '( x) x x 2 2 This could be rewritten as 1 1 x f '( x) 1 2 x 2x 2x 2 (but that’s not critical) (b) Find the slope of the tangent of f ( x ) when x=9. 1 Solution: f '(9) 6 1 So the slope when x=9 of the function is . 6 Easy (as usual) Example 4:(a) Find the derivative of f ( x ) 1 x Solution: f ( x) x 1 f '( x) x 11 x2 This could be rewritten as f '( x) 1 x2 (b) Find the slope of the tangent of f ( x ) when x=2. 1 Solution: f '(2) 4 1 So the slope when x=9 of the function is . 4 Easy (this is almost getting boring) Example 5:(a) Find the derivative of f ( x) 4 x 2 6 x 1 Solution: f ( x) 4 x2 6 x 2 3x2 1 f '( x) 8x 3x 2 6 x3 This could be rewritten as 3 x 6 f '( x) 8 x 3 x x Easy (Yawn) 3 x2