Problem 16 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\frac{1}{a}\left(x^{2}+\frac{1}{b} x+c\right) \quad(a, b, c \text { constant }) $$

Step-by-Step Solution

Answer

\( f'(x) = \frac{2x}{a} + \frac{1}{ab} \)

Step 1: Apply the Constant Multiple Rule

The function is given as \( f(x) = \frac{1}{a}(x^2 + \frac{1}{b} x + c) \). The constant multiple rule of differentiation states that the derivative of a constant multiple of a function is the constant multiplied by the derivative of the function. Therefore, we find the derivative: \( f'(x) = \frac{1}{a} \cdot \frac{d}{dx} \left(x^2 + \frac{1}{b} x + c\right) \).

Step 2: Differentiate Each Term

Now differentiate each term inside the brackets:\1. \( \frac{d}{dx} (x^2) = 2x \)\2. \( \frac{d}{dx} \left(\frac{1}{b} x\right) = \frac{1}{b} \)\3. \( \frac{d}{dx}(c) = 0 \) (since \(c\) is a constant).

Step 3: Combine the Derivatives

Combine these derivatives: \( \frac{d}{dx}(x^2 + \frac{1}{b} x + c) = 2x + \frac{1}{b} \).

Step 4: Multiply by the Constant Factor

Multiply the result from Step 3 by the constant factor \( \frac{1}{a} \): \( f'(x) = \frac{1}{a} (2x + \frac{1}{b}) \).

Step 5: Simplify the Expression

Simplify the expression by distributing the factor \( \frac{1}{a} \):\[ f'(x) = \frac{2x}{a} + \frac{1}{ab} \].

Key Concepts

Constant Multiple Rule

When dealing with differentiation, the Constant Multiple Rule is a handy tool. It says that if you have a constant multiplied by a function, you can take the derivative of just the function and then multiply the result by that same constant. For instance, if you have a function like \( f(x) = 3g(x) \), the derivative, \( f'(x) \), would be \( 3g'(x) \). In our original exercise, this means that the derivative of \( f(x) = \frac{1}{a}(x^2 + \frac{1}{b} x + c) \) starts by applying the constant \( \frac{1}{a} \) to the derivative of the inner function \( x^2 + \frac{1}{b} x + c \). It simplifies differentiation because it breaks down a more complex problem into easy steps.

Power Rule

The Power Rule is one of the most straightforward rules in differentiation and is widely used. If you have a term \( x^n \), where \( n \) is a constant, the Power Rule states that its derivative is \( nx^{n-1} \). This rule makes it easy to differentiate terms like \( x^2 \), which becomes \( 2x \).
In the original solution, we used the Power Rule to find the derivative of \( x^2 \), which was part of the polynomial function. This step is crucial because it helps you quickly and accurately determine how the polynomial function's slope changes at any point.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients, formed by the summation of terms that are each a constant raised to a power. Examples include expressions like \( x^2 + 2x + 3 \). Many functions, even complex ones, can be broken down into a polynomial form.
When you differentiate a polynomial, you apply differentiation rules like the Power Rule to each term separately. In our exercise, this was evident as we differentiated \( x^2 \), \( \frac{1}{b}x \), and \( c \). This systematic approach makes it manageable to work with polynomials and find their derivatives. By practicing with polynomial functions, you gain confidence in exploring more complex calculus problems.