Question

Compute the derivative of the given function. $$g(x)=14 x^{3}+7 x^{2}+11 x-29$$

Short answer

The derivative of the function is \( g'(x) = 42x^2 + 14x + 11 \).

Step-by-step solution

Identify the Terms in the Function

The function given is \( g(x) = 14x^3 + 7x^2 + 11x - 29 \). Notice that it is a polynomial with four terms: \( 14x^3 \), \( 7x^2 \), \( 11x \), and \( -29 \).

Differentiate Each Term Individually

For each term, use the power rule for differentiation, which states that the derivative of \( ax^n \) is \( anx^{n-1} \). Apply this to each term:- The derivative of \( 14x^3 \) is \( 42x^2 \).- The derivative of \( 7x^2 \) is \( 14x \).- The derivative of \( 11x \) is \( 11 \).- The derivative of \( -29 \), a constant, is \( 0 \).

Combine the Derivatives

Add the derivatives of each term to get the derivative of the entire function:\[ g'(x) = 42x^2 + 14x + 11 \]

Key concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, representing the rate of change of a function. Imagine you are watching a car on a straight road. If we graph its distance over time, the derivative tells us how quickly the car's position changes with respect to time. In simpler terms, it gives us the car's speed.
Similarly, for any function like height in relation to time or cost with respect to production, the derivative shows how one variable changes as the other changes.
Mathematically, if you have a function \( f(x) \), its derivative, denoted as \( f'(x) \) or \( \frac{df}{dx} \), indicates how \( f(x) \) changes as \( x \) changes. Calculating derivatives helps in understanding and predicting behavior in various real-world and theoretical scenarios.

The Power of the Power Rule

The power rule is a quick and easy tool used to find derivatives of polynomial functions. It works wonders, especially when dealing with terms where variables are raised to a power.
To use the power rule, consider a term of the form \( ax^n \). According to this rule, the derivative of \( ax^n \) is \( anx^{n-1} \). This means you multiply the coefficient \( a \) by the power \( n \), then decrease the power by one to find the new exponent.
This rule makes differentiation straightforward: no need for complex calculations, just a simple application of reducing the power by one and multiplying.

• For example, the derivative of \( 14x^3 \) is \( 42x^2 \), where you multiply 14 by 3 and reduce the power of 3 by one.
• The derivative of \( 7x^2 \) becomes \( 14x \), by multiplying 7 by 2 and reducing the power of 2 by one.

That's how seamless and effective the power rule can be for polynomial differentiation.

Polynomial Differentiation Explained

Polynomials are expressions made up of variables and coefficients combined through addition, subtraction, and multiplication, but no division by a variable. Differentiating polynomials involves applying the derivative rules to each term separately, making it a straightforward process.
In the function \( g(x) = 14x^3 + 7x^2 + 11x - 29 \), each term can be tackled individually. The power rule helps in this case, simplifying the differentiation process, as each term's exponent determines the application of the rule.

• Start with \( 14x^3 \), apply the power rule to get \( 42x^2 \).
• Next, with \( 7x^2 \), transform it into \( 14x \).
• For \( 11x \), since the power of \( x \) is 1, its derivative is simply 11.
• Lastly, any constant term, like \( -29 \), has a derivative of 0, because constants do not change.

When combined, these transformed terms provide the derivative \( g'(x) = 42x^2 + 14x + 11 \). Polynomial differentiation is systematic and precise, offering clarity to how functions change.