Question

Compute the derivative of the given function. $$f(x)=7 x^{2}-5 x+7$$

Short answer

The derivative of \( f(x) \) is \( f'(x) = 14x - 5 \).

Step-by-step solution

Understand the Problem

We need to find the derivative of the function \( f(x) = 7x^2 - 5x + 7 \). This requires using differentiation rules for each term in the function.

Identify Differentiation Rules

Recall that the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \), and the derivative of a constant is 0.

Differentiate the First Term

Apply the power rule to the first term \( 7x^2 \):\[ \frac{d}{dx}(7x^2) = 2 \times 7x^{2-1} = 14x \]

Differentiate the Second Term

Differentiate the second term \( -5x \) using the power rule:\[ \frac{d}{dx}(-5x) = -5 \cdot 1x^{1-1} = -5 \]

Differentiate the Third Term

Since the third term \( 7 \) is a constant, its derivative is 0:\[ \frac{d}{dx}(7) = 0 \]

Combine the Results

Combine the derivatives of each term to find the derivative of the entire function:\[ \frac{d}{dx}(f(x)) = 14x - 5 + 0 = 14x - 5 \]

State the Final Result

The derivative of the function \( f(x) = 7x^2 - 5x + 7 \) is \( f'(x) = 14x - 5 \).

Key concepts

Power Rule

The power rule is a fundamental tool in calculus, especially useful when dealing with polynomial functions. It provides a quick way to find derivatives of powers of a variable. In essence, the power rule states that \[ \frac{d}{dx}(x^n) = nx^{n-1} \].
This means that you multiply the original power by the coefficient and then reduce the power by one. For instance, when differentiating the term \( 7x^2 \) as in the original exercise, applying the power rule yielded \( 14x \).

• First, multiply the coefficient (7) with the power of \( x \) (which is 2).
• Decrease the power by 1 to get the new power of \( x \).
• The new term becomes \( 14x \), as calculated.

This simplification truly showcases the elegance and efficiency of the power rule, making it invaluable for calculus students.

Differentiation

Differentiation is the process of finding the derivative, which represents the rate of change of a function. It's a core idea in calculus used to understand how a function behaves. For any given function, its derivative can tell us things like slope at any point, or even the acceleration if applied in physics contexts.
To differentiate a whole function, break it down term by term, applying rules like the power rule as required. For example, in the function \( f(x) = 7x^2 - 5x + 7 \),

• The derivative of \( 7x^2 \) was found using the power rule.
• The derivative of \( -5x \) is \(-5\) since you apply the power rule to \( x^1 \), resulting in removing the variable.
• Lastly, the derivative of any constant term like 7, is 0.

Combine these results to find \( f'(x) = 14x - 5 \), which is the derivative of the entire function.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole number powers and a constant term. Examples include \( x^2 \), \( x^3 \), or linear terms like \( -5x \).
A polynomial function like \( f(x) = 7x^2 - 5x + 7 \) is made up of several terms, each differentiated separately.

• Start with the highest power term. Apply differentiation rules like the power rule to each part.
• Linear terms (like \(-5x\)) are derivative-friendly as they simplify to the original coefficient.
• Constant terms vanish because their rate of change is zero.

Understanding how to differentiate these functions is crucial, since polynomials frequently appear in both pure math and applied science situations. The process is methodical: focus on each term, and apply the rules you know. This ensures you capture the full behavior of the polynomial when differentiated.