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A curve has a horizontal tangent line at a point where the slope of the tangent, or the derivative of the function describing the curve, is zero. In simpler terms, this means the graph of the function flattens out at these points. Horizontal tangents often indicate local maxima, minima, or saddle points of a function.

To find where horizontal tangents occur, we first need to calculate the derivative of the function. Once we have that, we'll set the derivative equal to zero and solve for the values of \( x \). When you find these \( x \) values, you determine spots on the curve where the tangent does not rise or fall, which means it's horizontal.

Finally, substituting these \( x \)-values back into the original function gives us the corresponding \( y \)-coordinates to locate the exact points for horizontal tangents.

In calculus, a derivative represents the rate at which a function changes at any given point. It's analogous to finding the slope of a line but for a curve, which is more complex. Derivatives are essential for finding not only the slope at any point on a curve but also indicate how that slope changes, offering a look at the function’s behavior around specific points.

The symbol \( y' \) denotes the derivative of \( y \) with respect to \( x \). In our exercise, we started with \( y = 5x^{3} - 3x^{5} \), and the derivative, using calculus rules, simplifies intentionally complex expressions to something more manageable. By applying the Power Rule, we derived \( y' = 15x^{2} - 15x^{4} \).

This derivative helps us find where horizontal tangents occur by setting it to zero and solving for \( x \), which signify flat points along the curve.

Algebraic manipulation revolves around rearranging equations to isolate a particular variable, simplifying expressions, or factoring. This skill is crucial in calculus, as it allows us to handle complex derivatives and solve for desired quantities effectively.

During step 2 of the original exercise, we set the derivative equal to zero: \( 15x^{2} - 15x^{4} = 0 \). Simplifying this by factoring out common elements gives \( x^{2}(15 - 15x^{2}) = 0 \). Through factoring, we break down the equation into simpler parts. This makes it easier to find the roots of the equation or where the derivative equals zero.

The result, \( x = 0, 1, -1 \), provides potential points for horizontal tangents. Algebraic manipulation is thus an essential step to help solve equations efficiently.

The Power Rule is a fundamental rule in calculus that assists in differentiating functions of the form \( x^n \). The rule states: \( \frac{d}{dx}(x^n) = nx^{n-1} \). This simply means that you multiply the power by the coefficient and reduce the power by one.

In the exercise, we applied this rule to differentiate each term of the function \( y = 5x^{3} - 3x^{5} \). For \( 5x^{3} \), applying the Power Rule yields \( 15x^{2} \). For \( -3x^{5} \), it becomes \( -15x^{4} \). The Power Rule turns potentially complex derivatives into straightforward calculations.

Understanding and applying this rule allows for efficient handling of polynomial functions, a common type encountered in calculus. It’s a cornerstone in making the task of finding tangents or rates of change more accessible and manageable.