Find the equations of the lines tangent to the indicated...

find the equations of the lines tangent to the indicated curves at the given points.sketch the curve and tangent line. In Exercises 4 and $5,$ use the tangent feature of a calculator to view the curve and tangent line. $$y=\frac{1}{x^{2}+1} \text { at }\left(-1, \frac{1}{2}\right)$$

find the equations of the lines tangent to the indicated curves at the given points.sketch the curve and tangent line. In Exercises 4 and $5,$ use the tangent feature of a calculator to view the curve and tangent line.
$$y=\frac{1}{x^{2}+1} \text { at }\left(-1, \frac{1}{2}\right)$$

Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change at any point along its curve. It is obtained using differentiation techniques and provides the slope of the tangent line to the curve at a specific point, which is essential for understanding changes and approximations within the function’s behavior.

Tangent Line

A tangent line to a curve at a particular point is a straight line that touches the curve only at that point and shares the same slope as the curve at that point. It provides a linear approximation of the function near the point of tangency and offers insights into the local behavior of the function.

Point-Slope Form

The point-slope form of a line equation is a method used to express the equation of a line when the slope and one point on the line are known. This format is particularly useful for writing the equation of the tangent line, as it directly incorporates the slope obtained from the derivative and the coordinates of the point of tangency.

Chain Rule

The chain rule is a fundamental technique in differentiation used to compute the derivative of a composite function. When a function is expressed as a composition of two or more functions, the chain rule allows us to differentiate each layer and multiply their derivatives, which is crucial when dealing with functions that require a reorganization of terms, such as rewriting rational functions or applying exponent rules.

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