Combination Of Rules

3.18. Use whatever combination of rules is necessary- to find the derivatives of the following functions. Do not simplify the original functions first. They are deliberately kept simple to facilitate the practice of the rules.

Sx-2

The function involves a quotient with a product in the numerator. Hence both the quotient rule 8nd the product rule arc required. Stan with the quotient rule from (J.4).

7" mf where $(x) ■ 3x(2x - I). h(x) ■ Sx - 2. and h'(x) - 5. Then use the product rule from (J.J) for

¿'(a) -3t-2 + (2*-I)-3- I2X-3 Substitute the appropriate values in ihe quotient rule.

Simplify algebraically.

. 6QxJ-15x-24.r + 6-3ftcJ + 15* 3Qt* - 24* + 6 V " (Sx-2)1 " (5x - 2f

Note. To check this answer one could let y-ïx.— or y.— (2x-i) and use the product ruJe involving a quotient.

'Ilie function involves u product in which one function is raised to a power. Both the product rule and the generalized power function rule arc needed. Starting with the product rule, y'-*<*)• no+M*)f'<*)

Use the generalised power funcUoo rule for h'(x).

h'(x) = 2(4* - 5) 4 = H(4r - 5) = 32* - 40 Substitute the appropriate values in the product rule, y' - 3* (32* - 40) + (4x - 5)' • 3

and simplify algebraically.

y' - 9bx> - 12flx + 3(16^ - 40* + 25) - 144r - 240» + 75

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