**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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- Fall '07
- JonathanRogawski
- Math, Calculus, Exponential Function, Derivative, Implicit Differentiation, Power Rule