Lesson3.2 a basicdifferentiationrules
- 2. 2 Basic Derivatives • Constant function – Given f(x) = k • Then f’(x) = 0 • Power Function – Given f(x) = x n • Then =( ) 0 d k dx − = × 1 '( ) n f x n x ( ) − = × 1n nd x n x dx
- 3. 3 Try It Out • Use combinations of the two techniques to take derivatives of the following = 5 ( )f x x = 2 / 3 ( )g x x = 3 4 ( ) x h x x − = 3 ( )p x x
- 4. 4 Basic Rules • Constant multiple • Sum Rule • Difference Rule ( ) ( )× = ×( ) ( ) d d c f x c f x dx dx ( ) ( ) ( )+ = +( ) ( ) ( ) ( ) d d d f x g x f x g x dx dx dx ( ) ( ) ( )− = −( ) ( ) ( ) ( ) d d d f x g x f x g x dx dx dx How would you put these rules into words? How would you put these rules into words?
- 5. 5 Better Try This • Determine the following 2 2 3 ' ? y t t y = + − = ( ) 2 ( ) 3 5 '( ) ? f x x f x = − = 2 2 3 1 ( ) '( ) ? x x h x x h x − + = =
- 6. 6 An Experiment • Enter the difference quotient function into your calculator • Now graph the function and see if you recognize it ( ) ( )sin .01 sin .01 x x+ − Looks like the cosine function to me, pardner! Looks like the cosine function to me, pardner!
- 7. 7 Conclusion • We know that the limit of the difference function as h → 0 is the derivative • Thus it would appear that for f(x) = sin x f ‘ (x) = cos x • Make a similar experiment with the cosine function – What is your conclusion? ( )cos sin d x x dx = −
- 8. 8 Derivatives Involving sin, cos • Try the following 3 4 2cos sin 2 d x x dx π + − ÷ ( ) cos( ) '( ) ?f x x f xπ= × =
- 9. 9 Derivative Rule for ex • Experiment again … – Graph both – Make sure to set style on difference function to “Path” • What is your conclusion about ? .01 1 2 .01 x x x y e e e y + = − = xd e dx Let’s look at that Geogebra demo Let’s look at that Geogebra demo
- 10. 10 Try It Out • Find the derivative 3 2cos ' ? 4 x y e x y= + =
- 11. 11 Assignment • Lesson 3.2A • Page 136 • Exercises 1 – 65 EOO