91Ó°ÊÓ

When dealing with functions that are in the form of a ratio or fraction, the Quotient Rule is a vital tool in calculus for finding derivatives. Imagine we have two differentiable functions, say \( u(x) \) and \( v(x) \). If you want to find the derivative of a function \( y = \frac{u}{v} \), you will apply the quotient rule. This rule is expressed as:\[ y' = \frac{u'v - uv'}{v^{2}}\]In this expression, \( u' \) is the derivative of \( u \), and \( v' \) is the derivative of \( v \). This formula lets us effectively handle situations when the function's numerator and denominator are more complex expressions.

• First, differentiate \( u \) to find \( u' \).
• Then, differentiate \( v \) to find \( v' \).
• Use these derivatives to find \( y' \) using the formula.

For the function in our exercise, \( u = x^2 - 7x + 6 \) and \( v = x - 10 \). By applying the quotient rule, we obtain the derivative, helping us progress to finding critical points.

Once you have the derivative of a function, the next step is to find the Critical Points. These are the values of \( x \) where the derivative \( y' = 0 \) or where the derivative does not exist. Critical points are important as they can indicate where the function reaches maximum or minimum values.To find the critical points:

• Solve the equation \( y' = 0 \).
• Check for points where the derivative might be undefined (where the denominator of the derivative equals zero).

The critical points help in sketching the behavior of the function. They tell us where the function might change from increasing to decreasing, which is a hint toward local maxima or minima. In our specific function, solving \( y' = 0 \) gives potential turning points of the function. These are coupled with logical analysis of the domain and the behavior around these points.

After finding potential critical points, the next step is to determine whether these points are maximums, minimums, or something else. This is where the Derivative Test comes into play.The First Derivative Test can be used to determine if a critical point is a local maximum or minimum:

• If \( y' \) changes from positive to negative at a critical point, it's a local maximum.
• If \( y' \) changes from negative to positive, it's a local minimum.
• If \( y' \) does not change signs, the point is possibly a point of inflection.

The Second Derivative Test is another useful method:

• If the second derivative \( y'' > 0 \) at a critical point, the point is a local minimum.
• If \( y'' < 0 \), the point is a local maximum.
• If \( y'' = 0 \), the test is inconclusive.

Using these tests tells us not just where the critical points are but also their nature, helping us find the locations of maximum and minimum values of our function, rounding off the problem analysis.