L24 Math 233 Lecture Notes - Lecture 16: Differentiable Function
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Differentiability/differentials is differentiable at (a, b) if when we write x, b (a, ) y. + f y b f f y (x, ) (a. + + f lim y) Theorem: say f x and f y are defined on some disk containing (a,b) and continuous at. Given: (a,b), then f is differentiable at (a,b) (x, )dx. , write (x, )dy (x, ) y z = f y y z d = f x. The same idea without f(a,b) y = f (a, ) (x, ) Example: z = x2 + 3 y y z. Substitute (x-a) for dx and (y-b) for dy and (a,b) for (x,y) then we get l(x,y)-f(a,b) Assuming f and g are differentiable dz = dx dt dz dt dx. With two variables (t) y (x, ) x. Assuming all functions are differentiable, then z is a differentiable function of t and dz dz = dx dt dz dx + dt dt dy dt (t)
