ACTIVITY 5 IN DIFFERENTIAL CALCULUS Give the solution of derivative, the answer is on the right side. 5' If ft") = 4-\": 311d 3L") = 33': +1 Answer: 48.\\'{3x3 +1) nd If 8\"\") 3 1 (1'? s 6. If _1' = H" _9 and H = 3.\\" + 5 find 3 Answer: 12.\\'[3.\\" +5) 1- _ r '1 _ r 2 (it .e 7. If- (rJ) and. ((4): .nd (it Answer. 404) I 0'." 1 8. If 1' = 2)" + _1' + 4 . find Answer: \"'1' i 1s{4.\\-) Theorem 2.8. The Derivative of a Composite Function If the function g is differentiable at x and the function f is differentiable at g(x), then the composition function fog is differentiable at x and (fog) (x) = f(g(x))g'(x) Illustration 2.8. a) Let f(x)=x' and g(x)=x2 + 3x-1 find (fog) (x) Solution: Because f(x) = x then f'(x) =5x*, thus f(g(x)) = 58(x)]" fg(x))= 5(x2 +3x-1, furthermore Because g(x)= x +3x-1 then g'(x)=2x +3 Therefore (fog) (x)= f(g(x))g'(x) (fog) (x)= 5(x2 + 3x-1) (2x+3)Theorem 2.9. The Chain Rule of Differentiation If g is differentiable at x and if f is differentiable at g(x), then the composition fog is differentiable at x. Moreover, if y=f(g(x)) and u=g(x) then y=f(u) and dy _ dy du dx du dx In words, the derivative of a composite function of f and g is equal to the product of their respective derivatives. Illustration 2.10. dy a) Given y = u" and, u =2x' -5x +4 find dx Solution: from y = u" , the derivative is dy = 10u" du du from u =2x3 -5x +4, the derivative is dx - =6x- -10x, then dy dy du 10u (6x2 -10x), since u = 2x3 -5x +4, then dx du dx dy = 10(2x3 - 5x3 +4) (6x2 - 10x)\fTheorem 2.11. If the variable x is a function of y then the derivative dy dx is equal to the reciprocal of the derivative of the given function with respect to y . If x = g(y), then dy dx dx dy Illustration 2.12. a) Given x = 3y', find dy dx Solution: dx from x = 3y', the derivative is =9v dy dy V9x dx dx 912 , from x = 3y' the value of )= 3 dy dy dx 9v 9 V9x 181x2 9 3 9 dy 39x 3/9x dx 181x2 3/9x 729x3dy dx 9v 9x 9x 3 81x2 9 9 3 9 dy 39x 39x dx 3 81x2 39x 729x3 dy 9x dx 9x