Problem 6 - Calculus an Applied Approach

Question

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=\frac{1}{7}\left(5-6 x^{2}\right)} & {\left(1,-\frac{1}{7}\right)} \end{array}$$

Step-by-Step Solution

Answer

The derivative of the function at the point (1,-1/7) is -12/7. The Power Rule was used for differentiation.

Step 1: Identify the rule needed for differentiation

The function provided is \(f(x)=\frac{1}{7}(5-6x^2)\). Since it involves a power of x (specifically square of x), the Power Rule for differentiation is used. The Power Rule states that if a function is in the form of \(f(x)=ax^n\), its derivative is \(f'(x)=nax^{n-1}\).

Step 2: Apply Power Rule for differentiation

Applying this rule to the function \(f(x)\), we get \(f'(x)=\frac{1}{7}(0-2*6x)= -\frac{12x}{7}\). Here, we differentiated using the Power Rule, where \(n=2\), \(a=-6\) and got \(-2*-6x^{2-1}\), which simplifies to \(-\frac{12x}{7}\).

Step 3: Substitute the values of x into the derivative

We are given the point \((1,-\frac{1}{7})\), where x=1. Substituting \(x=1\) into the derivative, we get: \(f'(1)= -\frac{12(1)}{7}= -\frac{12}{7}\).

Key Concepts

Power Rule for differentiation

Understanding the Power Rule is crucial for students tackling calculus problems involving powers of variables. In essence, this rule simplifies the process of finding the derivative of monomial terms -- those with a base (variable) raised to an exponent. Specifically, the Power Rule states that if we have a function in the form \(f(x) = ax^n\), where \(a\) is a constant coefficient and \(n\) is a real number exponent, the derivative of that function with respect to \(x\) is \(f'(x) = nax^{n-1}\).

This rule quickly streamlines the differentiation process because it allows you to avoid using the definition of the derivative, which is more computationally intensive. When applying the Power Rule, it's important to first identify any coefficients and the exponent before carrying out the differentiation. If the function consists of multiple terms with different powers, apply the rule to each term separately.

Example of Using the Power Rule

For the function from our exercise \(f(x) = \frac{1}{7}(5 - 6x^2)\), focusing on the term with the variable, the Power Rule is implemented on \( -6x^2 \), resulting in \( -\frac{12x}{7} \) as the derivative.

Derivative of a function

When studying calculus, the derivative is one of the foundational concepts. It represents the rate at which a function is changing at any given point and is a pivotal tool for understanding the behavior of graphs and solving various practical problems in science and engineering. Essentially, a derivative provides a way to calculate the slope of the tangent line to the graph of the function at any point.

Conceptually, a derivative formula gives us the 'instantaneous' rate of change -- similar to how a speedometer shows the current speed of a car -- capturing how quickly the function's value is changing as its input changes. Derivatives have various notations, including \(f'(x)\), \(\frac{df}{dx}\), and \(\frac{d}{dx}[f(x)]\). Each of these notations expresses the derivative of the function \(f\) with respect to the variable \(x\).

Understanding through the Graph

If you visualize a curve on a graph, the derivative at a point tells you the slope of the line that just 'grazes' the curve at that point without cutting through it. This 'grazing' line is called the tangent line, and its slope is what we're calculating when we find the derivative.

Finding derivatives

The process of finding derivatives, often termed differentiation, involves a variety of techniques tailored to different types of functions. While the Power Rule is a quick method for basic polynomials, other rules such as the Product Rule, Quotient Rule, and Chain Rule are employed for more complex functions involving products, quotients, and compositions of functions, respectively. Additionally, trigonometric, exponential, and logarithmic functions each have their own derivative formulas.

To effectively find derivatives, one must first identify the structure of the given function and decide on the appropriate rule. Often, functions are composed of several terms that can each be differentiated independently, a property known as linearity of differentiation.

Practical Tips for Finding Derivatives

It's helpful to break down complex functions into simpler parts that can be differentiated using basic rules. Remember to simplify your function as much as possible before differentiating to make the process smoother. Moreover, keeping a list of common derivatives can serve as a quick reference to speed up differentiation.

Applying differentiation techniques

Applying differentiation techniques requires understanding and combining several rules of differentiation to tackle a wide array of functions. The differentiation process often starts with simplification where the function is broken down into its simplest parts. Students should be comfortable using basic rules like the Power Rule, as well as more advanced strategies like the Chain Rule for composite functions, or the Product and Quotient Rules for functions involving multiplication or division of terms.

When a function includes more than one term, such as in the exercise \(f(x) = \frac{1}{7}(5 - 6x^2)\), the derivative is found by differentiating each term individually. This is where the linearity property of derivatives comes into play, allowing the differentiation process to be applied term-by-term.

Chain of Differentiation

If the function comprises a composition of functions, the Chain Rule is applied, which involves taking the derivative of the outer function and multiplying by the derivative of the inner function.

A Systematic Approach

Another tip is to use a systematic approach: start by applying differentiation to the outermost operations (like exponentiation or trigonometric functions) and work inward, applying rules that match the current operation until the basic variables are reached.