y = 7x 2 + 2x 7 ( x, f (x)) y = 3x + 6 f (x) = 3( x 3) 2 dy dx = 3 dy dx =14x + 2 dy dy dx = 2x = 6x 18 dx dx = 2ax + b

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Download "y = 7x 2 + 2x 7 ( x, f (x)) y = 3x + 6 f (x) = 3( x 3) 2 dy dx = 3 dy dx =14x + 2 dy dy dx = 2x = 6x 18 dx dx = 2ax + b"

Sharlene Booker
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1 Rates of hange III Differentiation Workbook Limits For question, 1., draw up a artesian plane and plot your point [( x + h), f ( x + h) ] ( x, f (x)), and your point and visualise how the limit from first principles works as a simple rise over run gradient calculation, where the distance between the two points approaches zero. (Its not about getting an answer, its about visualising the situation so you understand what s happening.) 1. y = x $ For the remaining questions, just do the algebra to find the derivative of the given functions from first principles: 2. y = 3x y = 7x 2 + 2x 7 4. f (x) = 3( x 3) 2 5. y = ax $ + bx + c dx = 2x dx = 2ax + b dx = 3 dx =14x + 2 = 6x 18 dx Answers: For more practice, refer Exercise 1, Qu. 6 and 7.

2 Differentiate by Inspection 1. y = 4x 2. y = 4x y = 4x How can the derivative of qu. 1., 2. and 3. be the same? What does the derivative represent? Draw a diagram to show how the derivative of three different functions can be the same! 5. y = 3x 2 6. y = 7x 2 + 2x 7 7. f (x) = 3( x 3) 2 8. y = ax 2 + bx + c 9. Refer to your answers on the previous page for Qu. 3, 4 and 5. an you see the similarity to your answers? Are you surprised they are the same? Do you prefer to obtain the derivative from first principles, or by inspection? (Well you need to be able to do both!) f '(x) dx 10. When do you write and when do you write? f '(x) = 4 f '(x) = 4 f '(x) = 4 f '(x) = 6x f '(x) =14 x + 2 f '(x) = 6x 18 f '(x) = 2ax + b Answers: Tell Mr Finney your answer For more practice, refer Exercise 1, Qu. 8 to 19.

3 Differentiate by Rule the hain Rule 1. Expand the brackets and then differentiate y = ( x +1) 2 2. Use the chain rule to differentiate y = ( x +1) 2 3. Are you surprised you got the same answer? Which was easier? Although expanding brackets is sometimes quicker (as it was here), when functions become more complicated, the hain Rule will make differentiation much easier and quicker! 4. Practice using the hain Rule in this one: y = ( x +1) 3 5. In your answer to question 4, y'= 3 x +1, what exactly does this equation represent? It is obvious you can substitute values in for ( ) 2 x, but what does this give us? 6. ontinue to differentiate using the hain Rule: 1 y = 3x 2 7. f (x) = ( x 2 + 7x + 3) 3 dx = 2x + 2 f '(x) = 2x + 2 f '(x) = 3 x +1 3 f '(x) = 2 3x 2 Answers: Discuss your answer with Mr Finney f '(x) = 3( x 2 + 7x + 3) 2 ( 2x + 7) ( ) 3 ( ) 2 For more practice, refer Exercise 1D, Qu. 1 to 15.

4 Differentiate by Rule the Product Rule 1. Expand the brackets and then differentiate y = x 2 ( x +1) 2. Use the Product Rule to differentiate f (x) = x 2 ( x +1) 3. Are you surprised you got the same answer? Which was easier? Although expanding brackets is sometimes quicker (as it was here), when functions become more complicated, the Product Rule will make differentiation much easier and quicker! ontinue to practice using the Product Rule in these: 4. y = ( x +1) ( x 2 + 2) 5. y = 3x 2 ( x 3 + 2x 2 + 4) Now we can complicate things further by adding a power to the brackets. These are still Product Rule differentials, but you will also now need to use the hain Rule within the Product Rule process to solve these: 6. y = x( x +1) 2 7. f (x) = x 1 3x 2 For more practice, refer Exercise 1D, Qu. 16 & 17. And challenge yourself: 8. f (x) = ( x 2 + 3) ( 2x + 3) 2 dx = 3x 2 + 2x dx = 3x 2 + 2x dx = 3x 2 + 2x + 2 dx = 3x 5x 3 + 8x x Answers: No you were not surprised ( ) *+ = *, 3x$ + 4x f '(x) = = /,01 3x 2 2 $2(/,0$) ( 3x 2) dx =16x x x + 36

5 Differentiate by Rule the Quotient Rule 1. y = x +1 x f (x) = x 2 x f (x) = x x 3 For more practice, refer Exercise 1D, Qu. 18 to 21. Again, take care where there is a combination of rules required to evaluate the differential, such as here. 4. y = ( x + 2)2 x 5. ( )( x + 3) f (x) = x +1 x + 2 For more practice, refer Exercise 1D, Qu. 22. Enough of these easy ones. How about combining all three rules, in the one single differentiation process: 6. ( ) ( ) 2 y = x x 1 x +1 f '(x) = x( x + 2) ( ) 2 f '(x) = x ( x +1) 2 x 4 3x 1 = ( x + 2) 2 dx ( x +1) 3 dx = 1 x + 2 Answers: dx = x x 2 f '(x) = x 2 + 4x + 5

6 DERIVATIVES BY RULE What do you remember? In each of the following, determine the derivative by rule: YR 11 MATHEMATIS B TERM 3 1. f(x) = 3x 8 2. g(x) = 4x 11. T = $?, 5 3. y = : $ z$ 3z m = 2x(3x $ 4) 12. A car starts from rest and moves a distance, s metres, in t seconds, where s = : I t/ + : 1 t$. What is the initial acceleration and the acceleration when t=2? 5. y = (x 4)(x + 2) 6. f(x) =,5?$,, 5 7. p(t) = 4 t / 8. y = x $, D 9. f(x) = E:?F5 10. g(x) = 3 x(x $ + 2) ANSWERS: 1. f J (x) = 15x 1 2. g J (x) = 4 3. y J = z 3 *L 4. = *, 18x$ 8 5. y J = 2x 2 6. f J (x) = 1 7. p J (t) = 6 t 8. y J = : + I 8 2, O, O $ 9. f J (x) = : $2, D + : + / $ /, D $2, 10. g J (x) = :8 11. T J = ms, 2.5ms 0$

7 YR 11 MATHEMATIS B TERM 3 RATES OF HANGE II Determine the derivatives of: 1. f(x) = 3x / 2x $ + 5x y = 3x(x $ 2) 3. v = 8x g(x) = ax / + bx $ + cx + d 5. y = (2x 3) / 6. h(z) = 3z / 5z $ + 2z 5 + $ V 5 : 7. f(x) = /,D?$, 5?,, 8. y = (4x 3x $ + x / )(5x 3) 9. s(t) = 3t 01 6t 0$ D + 5 t + t 10. u = (x / ) y = x $ x / y = (3x $ 4x + 5) 1 X,D / 4Y/ 13. f(x) = (x $ 5)(4x + 3)(5x / + 13) 8 V D ANSWERS: 1. f J (x) = 9x $ 4x *+ *, = 9x$ 6 3. *[ *, = 8 4. g J = 3ax $ + 2bx + c 5. *+ *, = 24x$ 72x h J (z) = 9z $ 10z V D + / V O 7. f J (x) = 6x *+ *, = 20x/ 54x $ + 58x s J (t) = :$ ] + :$ ] D *^ *, = 12x:: + : D $ ] / ] y = 2x x / /,O $ D?: 12. *+ *, = 8(3x 2)(3x$ 4x + 5) / X,D / 4Y/ + 3x $ X,D / 4Y$ (3x $ 4x + 5) f J (x) = 2x(4x + 3)(5x / + 13) 8 + 4(x $ 5)(5x / + 13) x $ (5x / + 13) 1 (x $ 5)(4x + 3)

8 YR 12 MATHS B Term 1 DIFFERENTIATING RATIONAL FUNTIONS Differentiate each of the following with respect to x: 1. y = x 2. f(x) = 3 x 3. y = x O 4. g(x) = x / 5. y = $ 6. m = x x 7. y = /, 5 8. y =,? 9. h(x) = 2x x 10. y = x(1 x) 11. s = /0$, 12. y = $,0/?,5, ANSWERS: 1. y J = : $ 2. f (x) = / 3. y J = : 8 O $ 4. g (x) = / O 1 5. y J = : D 6. m J = / $ 7. y J = b 8 5 OR m J =, + x $ 8. y J = : $ 9. h (x) = 2 : $ 10. y J = :0/, $ 11. s = / : $ D 12. y J = : $ : D + b $2,

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,