Question

In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{f(x)=\frac{3}{x^{3}-4}} \quad \frac{\text { Point }}{\left(-1,-\frac{3}{5}\right)}\)

Short answer

The derivative of the function \(f(x)=\frac{3}{x^{3}-4}\) at the point \((-1,-\frac{3}{5})\) is \(-\frac{9}{25}\).\]

Step-by-step solution

Derivative calculation

To find the derivative of the function, employ the quotient rule. Let \(u = 3\), \(v = x^3 - 4\). The derivative of \(u\), denoted as \(u'\), is 0, since it is constant. The derivative of \(v\), denoted as \(v'\), is \(3x^2\), given \(v = x^3 - 4\). Substituting \(u\), \(v\), \(u'\) and \(v'\) into the quotient rule formula, the derivative of \(f(x)\), denoted as \(f'(x)\), is given by \[-\frac{3*3x^2}{(x^3 - 4)^2} = -\frac{9x^2}{(x^3 - 4)^2}.\] The negative sign before the fraction is due to the fact that the derivative of the numerator \(u\) is zero.

Evaluate the derivative at a point

Now, substitute \(x = -1\) into the derivative \(f'(x)\) to obtain the value of derivative at the point \((-1,-\frac{3}{5})\). Remember that the \(x\) value is -1. Hence, \(f'(-1) = -\frac{9*(-1)^2}{((-1)^3 - 4)^2} = -\frac{9}{(-5)^2} = -\frac{9}{25}.\]

Verify the result graphically

The derivative obtained corresponds to the tangent line slope at that point on the function graph. By graphing the function and visually confirming the slope at the point \((-1,-\frac{3}{5}\), the result can be verified. As such, you should expect a negative slope, steepening as the \(x\) value approaches -1.

Key concepts

Quotient Rule

The quotient rule is a crucial technique for calculus students learning how to differentiate functions that are presented as one function divided by another. To put it simply, when you have a function of the form \(f(x) = \frac{u(x)}{v(x)}\), you can compute its derivative by employing the quotient rule formula: \[f'(x) = \frac{u'v - uv'}{v^2}\].

Let's break down the steps you would typically follow:

• Identify the functions \( u(x) \) and \( v(x) \) which are, in our case, the numerator and the denominator of the given function.
• Find their derivatives separately, referred to as \(u'\) and \(v'\).
• Plug these into the quotient rule formula, being mindful of the signs as you combine the terms since subtraction is involved.

For students dealing with a constant as the numerator, like \(u = 3\) in our example problem, remember that the derivative of a constant is zero, which simplifies the computation. In essence, the quotient rule allows you to smoothly navigate the complexities of derivative calculation when division is in play.

Graphing Utility

Graphing utilities are powerful tools for understanding calculus concepts visually. These software programs or graphing calculators allow you to plot functions and their derivatives, providing an intuitive understanding of how the derivative represents the slope of the tangent line at any given point on the function's graph.

Here's how you can use a graphing utility to your advantage:

• Input the function provided in your exercise into the utility.
• Use the utility's feature to plot the derivative function you've calculated.
• Zoom in to the specific point of interest to see how the tangent line behaves.

By visually verifying the slope of the tangent line with a graphing utility, you'll not only confirm your calculated derivative at a specified point but also gain a deeper appreciation for the relationship between algebraic expressions and their graphical representations. This verification step is highly recommended to ensure that your calculations are correct, and it can also help you catch any errors you might have made.

Tangent Line Slope

The slope of the tangent line to a curve at a given point is a direct measure of how steep the curve is at that point, essentially determining the instantaneous rate of change. It's an essential concept in calculus, as it's synonymous with the derivative at that point. To find this slope, you would take the derivative of the function and evaluate it at your point of interest.

Here's how you can conceptualize this:

• The derivative function \(f'(x)\) provides a formula that calculates the slope of the tangent line for any value of \(x\) on the function \(f(x)\).
• Evaluating \(f'(x)\) at a given point gives you the specific slope of the tangent line at that \(x\) value.
• If the slope is positive, the function is increasing at that point, and if it's negative, the function is decreasing.

In our example problem, the negative value of the derivative at \(x = -1\) indicates a downward-sloping tangent line at that point on the curve. Understanding and calculating the slope of the tangent line is fundamental in detailing how a function behaves and changes at specific points, bridging the gap between abstract differentiation and observable graphical behavior.