Week 5 Course Notes - MATH 137 WEEK 5 NOTES PROF. DOUG PARK...

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MATH 137 WEEK 5 NOTES PROF. DOUG PARK 1. Chain Rule revisited Chain Rule . [ f ( g ( x ))] 0 = f 0 ( g ( x )) · g 0 ( x ) Remark . In Leibniz notation, if y = f ( u ) with u = g ( x ), then Chain Rule says dy dx = f 0 ( u ) · g 0 ( x ) = dy du · du dx . In other words, Chain Rule says du ’s “cancel” just like numbers. We will make use of this cancelation often in future applications. 2. Implicit differentiation This is a method of differentiation where we take derivatives of equations of the form f ( x,y ) = g ( x,y ). The crucial observation is that Chain Rule implies d dx = dy dx · d dy = du dx · d du = dx · d = ··· Ex . Find the equation for the tangent line to x 2 + y 2 = 25 at the point (3 , 4). Sol’n . Note that (3 , 4) lies on the circle of radius 5 centered at the origin. The equation for tangent is given by y - 4 = m ( x - 3) where m = dy dx ± ± ± ± (3 , 4) . (Here, vertical line means “evaluated at”.) We differentiate both sides of x 2 + y 2 = 25 using Chain Rule d dx = dy dx · d dy Date : October 16, 2009. 1
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2 DOUG PARK Taking x -derivatives of both sides, we get d dx ( x 2 + y 2 ) = d dx (25) 2 x + d dx ( y 2 ) = 0 2 x + dy dx · d dy ( y 2 ) = 0 2 x + dy dx · 2 y = 0 dy dx = - 2 x 2 y = - x y Hence we conclude dy dx ± ± ± ± (3 , 4) = - 3 4 . Equation for tangent line is y - 4 = - 3 4 ( x - 3) or y = - 3 4 x + 25 4 or 3 x + 4 y = 25. Remark . Note that the equation for the tangent line is gotten by replacing x 2 and y 2 in the equation for the circle by 3 x and 4 y , respectively. This is a general phenomenon and one can even prove the following theorem. Theorem
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Week 5 Course Notes - MATH 137 WEEK 5 NOTES PROF. DOUG PARK...

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