(a) find an equation of the tangent line to the graph of f...

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=\frac{1}{2} x \ln \left(x^{2}\right), \quad(-1,0) $$

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$$
f(x)=\frac{1}{2} x \ln \left(x^{2}\right), \quad(-1,0)
$$

Key Concepts

Derivative as Instantaneous Rate of Change

The derivative of a function at any point is defined as the rate at which the function is changing at that specific point. It is computed using the limit definition of the derivative and gives the slope of the tangent line to the function at that point, representing the best linear approximation to the function nearby.

Tangent Line

A tangent line to a function at a given point is a straight line that touches the function only at that local point and has the same slope as the function's derivative at that point. This concept is crucial in understanding linear approximations and the behavior of functions in the vicinity of that point.

Product Rule

The product rule is a fundamental rule in differentiation used to find the derivative of a product of two functions. It states that the derivative of the product is the derivative of the first function multiplied by the second, plus the first function multiplied by the derivative of the second. This rule is essential when the function in question is expressed as a product of simpler functions.

Chain Rule

The chain rule is used to differentiate composite functions, where one function is nested within another. It states that the derivative of the composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule enables simplification of derivatives in more complex functions.

Graphical Verification using Technology

Graphing utilities and computer algebra systems provide visual representations of functions along with their tangent lines, which helps in verifying analytical results. These tools often include features to compute derivatives directly, thereby offering a practical means to confirm the accuracy of calculated slopes and tangent lines.

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