Problem 14 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=3 x^{2}-5 x+4 $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = 3x^2 - 5x + 4 \) is \( f'(x) = 6x - 5 \).

Step 1: Identify the Function to Differentiate

We are given the function \( f(x) = 3x^2 - 5x + 4 \). Our goal is to find its derivative, which involves differentiating each term individually.

Step 2: Differentiate the First Term

The first term of the function is \( 3x^2 \). Using the power rule, which states \( \frac{d}{dx}(x^n) = nx^{n-1} \), the derivative of \( 3x^2 \) is \( 6x \).

Step 3: Differentiate the Second Term

The second term is \( -5x \). Using the rule for the derivative of \( ax^n \), where \( n = 1 \), we have that the derivative of \( -5x \) is \( -5 \).

Step 4: Differentiate the Constant Term

The constant term in the function is \( 4 \). The derivative of any constant is \( 0 \).

Step 5: Combine the Derivatives

Now, we combine the derivatives of each term: \( 6x \) (from \( 3x^2 \)), \(-5 \) (from \( -5x \)), and \( 0 \) (from the constant \( 4 \)). This gives us \( f'(x) = 6x - 5 \).

Key Concepts

Power Rule

The power rule is a fundamental tool in differentiation, used when dealing with functions in the form of polynomial expressions. It's particularly useful for terms that include a variable raised to a power, like our term \( 3x^2 \). The power rule allows us to succinctly find the derivative by following this simple formula: if you have a term \( x^n \), and you differentiate it with respect to \( x \), you get \( nx^{n-1} \). For our specific example, applying the power rule to \( 3x^2 \), we perform these steps:
- Multiply the exponent by the coefficient (2 by 3 in this case), which gives us 6.- Reduce the exponent by one (making it 1 from 2).
This results in the derivative \( 6x \). Such straightforward mechanics make the power rule a go-to technique for differentiating polynomial terms.

Derivative of a Constant

When it comes to constants, differentiation takes a very simple form. A key point to understand is that the derivative of a constant is always 0. This is because differentiation essentially measures how a function changes as its input changes, and a constant doesn’t change—it remains the same regardless of the input. Let's consider the constant term in our function, 4.
- Since 4 doesn’t change with respect to \( x \), its rate of change, and hence its derivative, is zero.- This principle applies to any constant in a function, making their derivatives consistently 0, simplifying our calculations significantly.
In conclusion, this rule allows us to efficiently handle constant terms in differentiation without any lengthy computations.

Function Differentiation

Function differentiation is the process of finding the derivative of a given function, which tells us how the function behaves as its inputs change. Each term in a function contributes something unique to the overall rate of change. So, understanding each part is key. When we differentiate a function like \( f(x) = 3x^2 - 5x + 4 \), here's what happens:
- We apply differentiation rules, like the power rule or the derivative of a constant across each term individually.- For terms involving the variable with an exponent, use the power rule: \( 3x^2 \) becomes \( 6x \).- For terms that have just the variable: \( -5x \) becomes \( -5 \), as it’s a linear term (where exponent is 1).- Constant terms drop out entirely, as their derivative is zero.
By systematically applying these rules, you can quickly reach the derivative \( f'(x) = 6x - 5 \), which represents the instantaneous rate of change of the original function at any point.