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Home Differentiation Chain Rule Chain Rule  Definition and Proof  
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Chain Rule  Definition and Proof
CHAIN RULE Let f, G be two real functions and define the new function g by the rule g(t) = G(f(t)). At any value of t where the derivatives f'(t) and G'(f(t)) exist, g'(t) also exists and has the value g'(t) = G'(f(t))f'(t). PROOF Let x = f(t), y = g(r), y = G(x). Take t as the independent variable, and let Δf ≠ 0 be infinitesimal. Form the corresponding increments Δx and Δy. By the Increment Theorem for x = f (t), Δx is infinitesimal. Using the Increment Theorem again but this time for y = G(x), we have Δy = G'(x) Δx + ε Δx or some infinitesimal ε. Dividing by Δt, Then taking standard parts, or


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