91Ó°ÊÓ

In calculus, a derivative represents the rate at which a function is changing at any given point, essentially describing the slope of the function's graph. The process of finding a derivative is known as differentiation. When you take the derivative of a function, you use notation like \(f'(x)\) or \(\frac{d}{dx}f(x)\) to indicate the derivative of the function \(f\) with respect to \(x\).

• The derivative provides a linear approximation to the function at a point, giving the slope of the tangent to the function's curve at that point.
• When the derivative is zero, the function has a horizontal tangent, often representing local maxima or minima.

For example, with the function \(f(x) = x^{2} - 6x + 5\), its derivative \(f'(x)\) is calculated as \(2x - 6\). This derivative can tell you how steep the function is rising or falling at any particular \(x\) value.

The power rule is a basic yet vital tool in calculus for finding derivatives of functions that are powers of \(x\). It states that the derivative of \(x^n\) (where \(n\) is any real number) is \(nx^{n-1}\). For instance, if you have a term like \(x^2\), the derivative would be \(2x^{2-1}\) or \(2x\). The power rule simplifies differentiation, particularly when dealing with polynomials.

• It helps break down complex expressions into manageable pieces by treating each term separately.
• Any constant factor, like \(-6\) in the term \(-6x\), is retained during derivation.

In the function \(f(x) = x^{2} - 6x + 5\), applying the power rule to each term results in the derivative \(2x - 6\). This quick computation technique is a staple for solving problems where derivatives are needed.

The slope of a curve refers to the steepness or incline of a curve at any given point. In calculus, the slope is essentially the derivative of the function at that point. Calculating the slope of a curve can provide insights into the behavior of functions and is pivotal in understanding motion, change, and shapes of graph lines.

• By setting the derivative equal to specific values, you can determine where the curve has particular slopes.
• For instance, determining when the slope is \(0\) helps find horizontal tangents, indicative of peaks or valleys in the curve.

In the exercise, determining the slope of the function \(f(x) = x^{2} - 6x + 5\) involves solving \(2x - 6\) both for when it equals \(0\) and \(2\). The solutions \(x = 3\) and \(x = 4\) describe positions on the curve where the tangent's slopes are \(0\) and \(2\), respectively, showcasing the dynamic nature of calculus and its utility in charting curves.