Consider the function

f(x) = { x² cos(2/x), x ≠ 0
0, x = 0

b. Determine f' for x ≠ 0.

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To find the derivative of the function f(x) = x² cos(2/x) for x ≠ 0, we will use the product rule. The product rule states that if you have a function h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x).

Identify the two functions: u(x) = x² and v(x) = cos(2/x). We need to find the derivatives of these functions separately.

First, find the derivative of u(x) = x². The derivative, u'(x), is 2x.

Next, find the derivative of v(x) = cos(2/x). Use the chain rule here. The chain rule states that if you have a composite function g(h(x)), then the derivative is g'(h(x)) * h'(x). Let g(t) = cos(t) and h(x) = 2/x. The derivative of g(t) is -sin(t), and the derivative of h(x) = 2/x is -2/x².

Apply the product rule: f'(x) = u'(x)v(x) + u(x)v'(x). Substitute the derivatives and functions: f'(x) = (2x)cos(2/x) + (x²)(-sin(2/x))(-2/x²). Simplify the expression to find f'(x) for x ≠ 0.

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Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. For a function f(x), the derivative f'(x) is defined as the limit of the average rate of change as the interval approaches zero. This concept is fundamental in calculus as it allows us to analyze the behavior of functions, including their slopes and rates of change.

Product Rule

The Product Rule is a formula used to differentiate products of two functions. If f(x) = u(x) * v(x), then the derivative f'(x) is given by f'(x) = u'(x)v(x) + u(x)v'(x). This rule is essential when dealing with functions that are products of simpler functions, such as the function f(x) = x² cos(2/x) in the given question.

Limit Definition of Derivative

The limit definition of the derivative states that the derivative of a function at a point is the limit of the difference quotient as the interval approaches zero. Mathematically, f'(a) = lim (h→0) [(f(a+h) - f(a))/h]. This definition is crucial for understanding how derivatives are derived and provides a foundation for more advanced concepts in calculus, especially when evaluating derivatives at points where the function is defined piecewise.

Consider the functionf(x) = { x² cos(2/x), x ≠ 00, x = 0b. Determine f' for x ≠ 0.