Suppose that the functions and g are defined throughout an open interval containing the point x 0 , that is differentiable at x 0 , that (x 0 ) 0, and that g is continuous at x 0 Show that the product g is differentiable at x 0 This process shows, for example, that although x is not differentiable ...

Lets prove that the product of two functions x and gx is differentiable at a point x0 given that x0 ...

Suppose that the functions and g are defined throughout an open interval containing the point x

Question:

Suppose that the functions ƒ and g are defined throughout an open interval containing the point x₀, that ƒ is differentiable at x₀, that ƒ(x₀) = 0, and that g is continuous at x₀. Show that the product ƒg is differentiable at x₀. This process shows, for example, that although|x| is not differentiable at x = 0, the product x |x | is differentiable at x = 0.

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