Find the equation of the line tangent to the function at the...

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Key Concepts

Point-Slope Form

The point-slope form is a way to express the equation of a line when you know its slope and a point through which it passes. This form is given by the equation y - y1 = m(x - x1), where m represents the slope and (x1, y1) is the point on the line. It is particularly useful for writing the equation of a tangent line after computing the slope using the derivative and evaluating the function at the given point.

Derivative

The derivative of a function is a fundamental concept in calculus that represents the instantaneous rate of change of the function with respect to its variable. It is calculated using differentiation, which can be applied using rules such as the power rule or the limit definition of the derivative. In the context of finding a tangent line, the derivative evaluated at a particular point gives the slope of the tangent line at that point.

Tangent Line

A tangent line to a function at a given point is a straight line that touches the graph of the function only at that point (locally) and has the same slope as the function does at that point. The concept of a tangent line is central in calculus, as it provides a linear approximation of the function near the point of tangency, which is useful in various applications such as solving differential equations and optimization problems.

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