Questions, with answers, explanations and proofs, on derivatives of even and odd functions are presented.

Assume that function f is differentiable everywhere, which of the graphs A), B), C) or D) is the graph of the first derivative of f? The given function is even, henceSolution to Question 1:f(x) = f(-x) Differentiate the two sides of the above equaltion. df/dx = d(f(-x))/dx To differentiate f(-x), we use the chain rule formula as follows: Let u = - x, hence d(f(-x))/dx = df(u)/du . du/dx = f '(u) . (-1) = - f '(-x) Substituting in df/dx = d(f(-x))/dx, we obtain f '(x) = - f '(- x). f '(- x) = - f '(x) and therefore this is the proof that the derivative of an even function is an odd function. Analyzing the 4 graphs A), B), C) and D), only A) and B) are odd. Analyzing the graph of f; f is a decreasing function from the maximum on the left to the origin (0,0). Hence in this interval the derivative must be negative. Graph B) fulfills this condition and therefore the answer is B)
Assume that function f is differentiable everywhere, which of the graphs A), B), C) or D) is the graph of the first derivative of f?
The given function is odd, hence
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