Question

Use implicit differentiation to find the following. (Round answers to four decimal places as needed. If only th (xy)2 + xy - x = 3,(-3,0) (a) the expression of the slope of the tangent line in terms of x and y dy. -2012 – y + 1 dx2xy + x (b) the equation of the tangent line at the indicated point on the graph y =

Answer 1

☺ @ Gwen (ay)° +xy - x = 3, (3, 0) = x@y? + xy -X =3 differentialing wort. x, we get 2x? y dy + 2xy2 + x dy +y +=0 (184 +9) de des232.9+ da 2xży te , - . Equation of tangent line at (-3.0) is, 4-0 = -4 (x+3) y = لنز مي 7 @ Given ln (x-4) +2 = 2x² , x=0 so after differentiating word o, we get Acy ( de Icoane = que 4x² - 4x4 →

- 4x + 484 + at to, An(.y) = 822-1) m-y = 62(x+1) 3 ya x-e? 5 ye 0-8761) = 42 so 3. Equation of tangent is 9 + 1 = 1 (2-0) arven fig = x² = y (say ) and . Equation of y'a Tangent ba? , S(:1) = 3. line il 5 9 : 13x17]

( | ROR xt 3 | SI-2 Gven fra) = X ,=2 | so y = ft) - 2+ 그 C and f(x) = 1 - = 1 - 4 so the equation of fangent line is, y - k = + (2) | 9 -3 = | → 3 = 4 글 + = +3 • 3 1 +3 .

Answer 2

ac y - x = 3 0 (0) (2+ -4-5, (-3,0) (0) (ayyºr ay - X=3 > x*y? + xy - x = 3 O The expression of the slope of the tangent line is differentiation of y with respect to x. Diferentiate 0 with respect to lace you) + (27- () = (3) - I = 0 os termeny + y(22)2 419+ y** -1-0 + gx ya txxl ay ty-1=0 = o . d dx I dx yata * X XI du dx Scanned with CamScanner

dx > day 9 + day' + 20y + y-t=0 > (22'yt a ) + dayty-two => (da?y + x) dx = -dxy'-y #1 (6-) line at the The equation of the tangent point (%, Ye) is y - Yo = m (2-2) where m= dy | da lat (22, Yz) Here ( 4 , Yu) = (-3,0) (3-0) = m ( x - (-3)) y = m (x+3) Scanned with CamScanner

m=) at(-1,0) m = V +1 + 2 lat (-3,0) -3 x 3)xo* - 0 +1 2(-3)*x0 + 6-3) 0-0 + 1 (3) - Scanned with CamScanner