Tiêu đề Tutorial5 Final MAT183.pdf ✅

Tutorial 5: Implicit Differentiation MAT183 (Calculus I) Mahani@Mar25 27 9. Use the first principle of differentiation to find f ′ (x) for the function f(x) = 2x 3−x . [Oct08] 6 (3 − x) 2 10. Find dy dx for xy = ln(cos x + sin y) by using implicit differentiation. [Apr09] y + sin x cos x + sin y cos y cos x + sin y − x 11. Find dy dx for x 2e 4x+y = sin2 (3x) + ln(y 2 ) by using implicit differentiation. [Oct09] 6 sin(3x) cos(3x) − 2xe 4x+y − 4x 2e 4x+y x 2e 4x+y − 2 y 12. Find dy dx for 2x 2y 2+e x 2y 2 = sec(x + 1) by using implicit differentiation. [Apr10] sec(x + 1) tan(x + 1) − 4xy 2 − 2xy 2e x 2y 2 4x 2y + 2x 2ye x 2y2 13. Find dy dx for 3xe −y + y 2 sin x = xy by using implicit differentiation. [Oct10] y − 3e −y − y 2 cos x −3xe −y + 2ysin x − x 14. Find dy dx for 5 cos(x + y) − 3e y = 4x 2 by using implicit differentiation. [Apr11] 8x + 5 sin(x + y) −5 sin(x + y) − 3e y 15. Find dy dx for e 2x−3y = ln(xy 4 ) by using implicit differentiation. [Sep11] 1 x − 2e 2x−3y −2e 2x−3y − 4 y 16. Find dy dx for sin(x + y) = x 6 + 3xy by using implicit differentiation. [Mar12] 6x 5 + 3y − cos(x + y) cos(x + y) − 3x
Tutorial 5: Implicit Differentiation MAT183 (Calculus I) Mahani@Mar25 28 17. Find dy dx for x 3y = cos(e y ) by using implicit differentiation. [Oct12] −3x 2y x 3 + e y sin(e y) 18. Find dy dx for 6 + sin(2y) = 2y 3 + 3x by using implicit differentiation. Hence, find the equation of the tangent line to the curve 6 + sin(2y) = 2y 3 + 3x at the point (2, 0). [Mar13] 3 2 cos(2y) − 6 y 2 , y = 3 2 x − 3 19. Find dy dx for tan2 y − 4e 8x = ysec x by using implicit differentiation. [Sep13] ysecxtanx + 32e 8x 2 tan y sec2x − sec x 20. Find dy dx for y 3 + 2xy − x 4 = cos x by using implicit differentiation. [Mar14] 4x 3 − 2y − sin x 3y 2 + 2x 21. Find dy dx for e 4x+7y = x 2 − ln (xy 3 ) by using implicit differentiation. [Sep14] 2x − 1 x − 4e 4x+7y 7e 4x+7y + 3 y 22. Find dy dx for y 3 + 4xln(2y) = e 2y + sin2x by using implicit differentiation. [Mar15] 2 sin xcosx − 4ln(2y) 3y 2 + 4x y − 2e 2y 23. Find dy dx for 2yln(x 2 ) + 3x 2 = 2 tan(x − y) − e x 2 by using implicit differentiation. [Sep15] 2sec2 (x − y) − 2xe x 2 − 4y x − 6x 2ln(x 2) + 2sec2(x − y) 24. Find dy dx for yln(x 4 ) + cos(2y) = x 3y 2 by using implicit differentiation. [Mar16] 3x 2y 2 − 4y x ln(x 4) − 2 sin(2y) − 2x 3y
Tutorial 5: Implicit Differentiation MAT183 (Calculus I) Mahani@Mar25 29 25. Find dy dx for 3y 2x + 7e x ln(sin y) = 4x 2 by using implicit differentiation. [Oct16] 8x − 7e x ln(sin y) + 3y 2x 2 3 2x + 7e x cot y 26. Find dy dx for ln(sin(y 2 )) − x y = e 3y by using implicit differentiation. [Mar17] 1 y 2ycot(y 2) + x y 2 − 3e 3y 27. Given the function ye x−1 + cos(3y) = 2 − x 2 . [Jan18] a) Find dy dx using implicit differentiation. ye x−1 + 2x 3 sin(3y) − e x−1 b) Find the equation of the tangent line to the function at the point (1,0). y = −2x + 2 28. Given the function ln(cos( 2y) + ye 3x−6 +x 2 = 4. [Jun18] a) Find dy dx using implicit differentiation. 3ye 3x−6 + 2x 2 tan(2y) − e 3x−6 b) Find the equation of the tangent line to the function at the point (2,0). y = −4x + 8 29. Given the function 4 ye 2x + sin(xy) = ln(2x − 1) . [Dec18] a) Find dy dx using implicit differentiation. 2 2x − 1 − 8ye 2x − ycos(xy) 4e 2x + xcos(xy) b) Find the gradient of the tangent line to the function at the point (1,0). 2 4e 2 + 1 30. Given the function ln(2 − e 1−x ) + xy 2 + 1 = 2x x+1 + 3y. [Jun19] a) Find dy dx using implicit differentiation. 2 (x + 1) 2 − e 1−x 2 − e 1−x − y 2 2xy − 3 b) Find the equation of the tangent line to the function at the point (1,3).