Find f^'(x) and the equation of the line tangent to the...

Find $f^{\prime}(x)$ and the equation of the line tangent to the graph off at the indicated value of $x .$ Find the value(s) of $x$ where the tangent line is horizontal.
$$
f(x)=(2 x-1)^{3} ; \quad x=1
$$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function is changing at any point. It forms a core concept in calculus and is used to analyze functions by calculating slopes, rates of change, and other dynamic properties.

Chain Rule

The chain rule is a fundamental technique in differentiation used when dealing with composite functions, where one function is nested inside another. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Tangent Line Equation

The tangent line equation represents the line that just touches a curve at a given point and has the same slope as the function at that point. It is used to approximate the function locally and is typically found using the point-slope form of a line by incorporating the function’s derivative as the slope.

Horizontal Tangent Lines

A horizontal tangent line occurs when the derivative of the function is zero, indicating that the slope at that point is flat. Finding points where the derivative equals zero helps identify local extrema and critical points where the function changes from increasing to decreasing or vice versa.

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