Problem 108 - Calculus Volume 1

Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=4 x^{2}-7 x $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 8x - 7 \).

Step 1: Identify the Function

The function given is a polynomial function: \[ f(x)=4x^{2}-7x \]. This function is a combination of two terms: one quadratic term \( 4x^2 \) and a linear term \( -7x \).

Step 2: Apply Power Rule to First Term

To differentiate \( 4x^2 \), apply the power rule, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).Here, differentiate \( 4x^2 \): \[ \frac{d}{dx} [4x^2] = 4 imes 2x^{2-1} = 8x \].

Step 3: Apply Power Rule to Second Term

For the term \( -7x \), also use the power rule. Realize \( -7x \) is equivalent to \( -7x^1 \), thus by applying the rule:\[ \frac{d}{dx} [-7x] = -7 \times 1x^{1-1} = -7 \].

Step 4: Sum the Derivatives

Finally, combine the derivatives of each term to find the overall derivative of the function:\[ f^{ ext{'} }(x) = 8x - 7 \].

Step 5: State the Final Derivative

Therefore, the derivative of the function is: \[ f^{ ext{'} }(x) = 8x - 7 \].

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus, particularly useful when differentiating polynomial functions. This rule provides a straightforward way to find the derivative of any basic power of a variable. If you have a function defined as \( f(x) = ax^n \), the power rule states that its derivative is \( f'(x) = anx^{n-1} \). In simple terms, it involves multiplying the entire term by the exponent, and then reducing the exponent by one.

Let's apply this rule to a couple of examples:

• For \( 4x^2 \), multiply the term by the exponent (which is 2), resulting in \( 4 \times 2 \), and decrease the exponent by 1, ending up with \( 8x^{1} \), or simply \( 8x \).
• In the case of \(-7x \), think of it as \(-7x^1 \). Multiply \(-7\) by the exponent \(1\), and reduce the power to get \(-7 \times 1x^{0} \), simplifying to \(-7\).

This approach enables the differentiation of each term individually, which can then be combined to find the overall derivative of the function. It's a powerful tool for handling polynomial functions efficiently.

Polynomial Function

A polynomial function is a type of function that is expressed as a sum of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. For example, \( f(x) = 4x^2 - 7x \) is a polynomial function with two terms: \( 4x^2 \) (a quadratic term) and \(-7x \) (a linear term).

Polynomial functions are easy to work with when performing differentiation, thanks to rules like the power rule. They can take various forms, boasting tremendous flexibility and usability in mathematical modeling and problem-solving. Some features of polynomial functions include:

• They have non-negative integer exponents.
• The coefficients can be any real number, positive or negative.
• They are defined for all real numbers.

Being able to identify and manipulate polynomial functions is key in calculus, as they often form the foundation for more complex problems and solutions.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. In other words, it tells us how much a function \( f(x) \) changes as \( x \) changes. This is represented mathematically as \( f'(x) \).

For polynomial functions, differentiation involves applying the power rule to each term in the polynomial separately and then combining the results. The derivative of a function provides insights into several important properties:

• It helps determine the slope of the function at any given point.
• It is useful in finding maxima and minima, indicating points where the function may have peaks or troughs.
• It assists in understanding the concavity or curvature of a function across its domain.

By practicing differentiation, you gain a deeper understanding of how functions behave. In our case with the function \( f(x) = 4x^2 - 7x \), distinguishing each term individually using the power rule allows us to arrive at the simple derivative \( f'(x) = 8x - 7 \). This tells us how the function's output changes in relation to small changes in \( x \).