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Let g1(x), ... , gn(x) be differentiable functions. If f(x) = g1(x)...gn(x), prove that its derivative is

f'(x) = SUMMATION[i=1,n] (f(x) gi'(x)/gi(x)) .

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Differentiable functions are exemplified in this solution.

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Let be differentiable functions. If prove that its derivative is

Proof: Let f(x)=g_1 (x)⋯g_n (x) be a product of differentiable functions g_1 (x),...,g_n (x) and let P(n) denote the statement
P(n): f^' (x)=∑_(i=1)^n▒(f(x) g_i^' (x))/(g_i (x))
Now if f(x)=g_1 (x), then
f^' (x)=g_1^' ...

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