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

(a)Find an equation of the tangent line to the graph of the function 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. $$ f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x \quad(-1,2) $$

(a)Find an equation of the tangent line to the graph of the function 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.
$$
f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x \quad(-1,2)
$$

Key Concepts

Tangent Line

The tangent line to a curve at a particular point is the straight line that has the same slope as the curve at that point. It serves as the best linear approximation of the function near the point of tangency and is central to understanding instantaneous change in calculus.

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. It is used to find the slope of the tangent line at any given point and is a fundamental concept in differential calculus.

Chain Rule

The chain rule is a method for differentiating composite functions. When a function is composed of other functions, the chain rule provides a systematic way of calculating its derivative by multiplying the derivatives of the outer and inner functions.

Point-Slope Form

The point-slope form is a way of writing the equation of a line when a point on the line and its slope are known. This form is particularly useful in problems where the task is to find the equation of the tangent line, as it directly incorporates both the slope (given by the derivative) and the coordinates of the point of tangency.

Graphing Utility and Verification Techniques

Graphing utilities are tools that allow for visualizing functions and their tangent lines, providing both a graphical confirmation of analytical work and a means to explore the behavior of functions. Additionally, many of these tools include features to calculate derivatives directly, which can be used to verify the results obtained through manual differentiation.

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