MAT 21C Lecture Notes - Lecture 26: Differentiable Function, List Of Trigonometric Identities
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The case that we can apply the chain rule for functions of two variables is when: X = x(t) -- x is a differentiable function of t. Y = y(t) -- y is a differentiable function of t. In order to apply the chain rule, we observe the following theorem: Dw/dt = f x (x(t), y(t))x"(t) + f y (x(t), y(t))y"(t) A more general way of phrasing this is: dw/dt = df/dx * dx/dt + df/dy * dy/dt. ** professor showed a quick proof of this to the class, can be found online ** Example: w = xy, find w" at t = pi/2 where x = cos(t) and y = sin(t) Then we proceed to plug in the values of x and y into this: = (sin(t) * -sin(t)) + (cos(t) * cos(t)) The same structure as above can be extrapolated to include for cases where we have three intermediate variables and one independent variable:
