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definition of derivative

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f(x)={(x^3)cos(1/x) if xâ? 0, 0 if x=0,
g(x)={(1/x)sin(x) if xâ? 0, 0 if x=0.

a) Using the definition of the derivative show that f is differentiable at 0 and determine f '(0).
b) Is g differentiable at 0? Justify your answer.
c) Show that f ' ^(x) and g ' ^(x) exist for xâ? 0 and determine their values.

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Solution Summary

This solution shows how to use the definition of derivative to show that given variables are or are not differentiable at 0.

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Definition of the Derivative

The derivative of a continuous function at x is the slope of the tangent line to the curve at x. The attached pdf file develops the idea of a derivative first using slopes of secant lines and then introducing and explaining the difference quotient in detail. An example and an explanation are provided for using the limit of the difference quotient to find the derivative of the function f(x) = x^2 + 4x.

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