MATH116 Lecture Notes - Lecture 14: Quotient Rule, Implicit Function

• School: University of Waterloo
• Department: Mathematics
• Course: MATH116
• Professor: Fiona Dunbar

Document Summary

Tutorial solutions 5: find dy/dx by implicit di erentiation. x + y = 1 + x2y2. Y(cid:48)(cid:18)1 (x + y) 1/2(1 + y(cid:48)) = 2xy2 + x2(2yy(cid:48)) 1. 2 (x + y) 1/2y(cid:48) 2x2yy(cid:48) = 2xy2 1. 1 4x2y x + y 1 x + y y(cid:48) : find y(cid:48)(cid:48) by implicit di erentiation. y2 + 3y = x. 2yy(cid:48) + 3y(cid:48) = 1 y(cid:48)(2y + 3) = 1 y(cid:48) = Now we can di erentiate this equation again to nd y(cid:48)(cid:48): y(cid:48)(cid:48) = (2y + 3) 2(2y(cid:48)) = 2 (2y + 3)3: find the derivative of the function. Do not simplify. (a) g(t) = sin 1(ln(t)) (b) h(z) = tan 1(cos( z2 + 2)) 1(cid:112)1 (ln(t))2 (cid:18)1 (cid:19) t (a) g(cid:48)(t) = (b) h(cid:48)(z) = 1 z2+2 (2z): use logarithmic di erentiation to nd the derivative of the following function. x3ex(x3 + 1)9 y = Solution: ln(y) = ln( y y(cid:48) = 3.