Question

Differentiate the function using the definition and find the slope of the tangent line at the given value of the independent variable 1) g(x) Find an equation of the tangent line and the normal line at the indicated point on the graph of the function. Use the definition of the derivative find the slope. 2) w = g(2) = 1/4 -2, (7,W) =(3,1) Find the first and second second derivative. 3) w=2-4-

Answer 1

(2 we have , g(x) ata Let h yo be small very very Now, 8 gath)--g(a) 9txth gta (9+2-9-8-2) (94x+h)(9+x) 8 h (9+x+h) (9+80) g Cath) – g(x) 8 h 6+2+h) (9+) o's by definition, 1 8 lim ho h 670 94x+h) (94) OP g'(a) = len g(x+h) – g(x) - (9+x2 zien, gʻcm) slope of the (9+2)2 tangent (9+3) line at x=3 is 8 = = g (3) - ne (ans). 144

be (2) We have, w = g(Z) = 4-7 Let hyo small. very very Now gCzth) - gez) = √4 -Z-h - V4-7 4-z-h-4+ Z S4-z-h +√4 Z h JH-Z-h+VA-Z So by definition of derivative we have g'cz) = lim g(x+h)- gcz) h to h lim hto SA-Z-h+JA-Z 1 122 h +972) 11 1 2 SHZ+√A-Z 2/A-Z . Slope of the tongent line at (2, 2) = (3,1) 2/A-3 @ns) is é

we have geven, W = g(t) = √A-Z -> ☺ Differentiating both sides woroto z we z get, do IZ 2/4-Z Noo do 7 dz (3,1) 2/A-3 & the eqreation of the tangent line to & at (3, 1) (0-1) do Can (2-3) (3,1) OP, 2-1 = -1 (2-3) QW-2=-zt 3. Ory Z +2w Ats0 the equation of the normal to at (31) 1 (6-1) (2-3) diw is OPO - d Z J (311) Or, OP, W-1 - 12 (2-3) 2-1 27-6. 2z - W=5 (ans) 0709

(3) we have, Z E + -425 duo do Z? Cans) 5 -6 = 207 2 do 3 Z ans)