121616949-math.101 - 4.9 Implicit Differentiation 87 factor...

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4.9 Implicit Differentiation 87 factor out the y , just as in the previous example. If you ever get anything more difficult you have made a mistake and should fix it before trying to continue. It is sometimes the case that a situation leads naturally to an equation that defines a function implicitly. EXAMPLE 4.18 Consider all the points ( x, y ) that have the property that the distance from ( x, y ) to ( x 1 , y 1 ) plus the distance from ( x, y ) to ( x 2 , y 2 ) is 2 a ( a is some constant). These points form an ellipse, which like a circle is not a function but can viewed as two functions pasted together. Because we know how to write down the distance between two points, we can write down an implicit equation for the ellipse:
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Unformatted text preview: 2 + p ( x-x 2 ) 2 + ( y-y 2 ) 2 = 2 a. Then we can use implicit differentiation to find the slope of the ellipse at any point. EXAMPLE 4.19 We have already justified the power rule by using the exponential function, but we could also do it for rational exponents by using implicit differentiation. Suppose that y = x m/n , where m and n are positive integers. We can write this implicitly as y n = x m , then because we justified the power rule for integers, we can take the derivative of each side: ny n-1 y ± = mx m-1 y ± = m n x m-1 y n-1 y ± = m n x m-1 ( x m/n ) n-1 y ± = m n x m-1-( m/n )( n-1) y ± = m n x m-1-m +( m/n ) y ± = m n x ( m/n )-1...
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