14.4 Notes - Section 14.4 The Chain Rule Shubhodeep...

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Section 14.4 The Chain Rule Shubhodeep Mukherji Chain Rule for a single variable, we said that when w = f ( x ) was a differentiable function of x and x = g ( t ) was a differentiable function of t , w became a differentiable function of t and dw dt could be calculated with the formula dw dt = dw dx dx dt . For functions of two or more variables the Chain Rule has several forms. The form depends on how many variables are involved but works like Chain Rule for a single variable after accounting for pres- ence of additional variables. 1 Functions of Two Variables If w = f ( x,y ) has continuous partial derivatives f x and n f y and if x = x ( t ), y = y ( t ) are differentiable functions of t , then the composite w = f ( x ( t ) ,y ( t )) is a differentiable function of t and df dt = f x ( x ( t ) ,y ( t )) * x 0 ( t ) + f y ( x ( t ) ,y ( t )) * y 0 ( t ) , or dw dt = ∂f ∂x dx dt + ∂f ∂y dy dt Example 1 Applying the Chain Rule Find the derivative of w = xy with respect to t along the path x = cos ( t ) , y = sin ( t ) . Solution 1
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dw dt = ∂w ∂x dx dt + ∂w ∂y dy dt = ( xy ) ∂x d dt ( cos ( t )) + ( xy ) ∂y d dt ( sin ( t )) = y ( - sin ( t )) + x ( cos ( t )) = ( sin ( t ))( - sin
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14.4 Notes - Section 14.4 The Chain Rule Shubhodeep...

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