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**Unformatted text preview: **Lecture 8: The Chain Rule May 15, 2009 Lecture 8: The Chain Rule Objectives 1 Use the generalized chain rule to find the partial derivatives of multivariable functions that depend on other multivarible functions. 2 Solve related rates problems. 3 Use implicit differentiation to compute partial derivatives Lecture 8: The Chain Rule Chain rule: Special case 1 Suppose that your path is given by the parametric equations x ( t ) and y ( t ), where t is time. You are travelling on a mountain with elevation z = f ( x , y ). Question: How can you find your rate of change of elevation with respect to time? ( dz dt ) Lecture 8: The Chain Rule Chain rule: Special case 1 Suppose that your path is given by the parametric equations x ( t ) and y ( t ), where t is time. You are travelling on a mountain with elevation z = f ( x , y ). Question: How can you find your rate of change of elevation with respect to time? ( dz dt ) Answer: Chain rule dz dt = ∂ z ∂ x dx dt + ∂ z ∂ y dy dt Lecture 8: The Chain Rule Class exercise Suppose z = p x 2 + y 2 and x ( t ) = e t , y ( t ) = 1- e t . Find dz dt Lecture 8: The Chain Rule Class exercise...

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