Problem 31 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$Q=a P^{2}+b P^{3}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dQ}{dP} = 2aP + 3bP^2 \).

Step 1: Identify the Function

The function we need to differentiate is \( Q = aP^2 + bP^3 \). This function is a polynomial in terms of \( P \) with constants \( a \) and \( b \).

Step 2: Apply the Power Rule for Derivatives

Recall that the power rule states that if \( f(P) = P^n \), then the derivative \( f'(P) = nP^{n-1} \). We will apply the power rule to each term of the polynomial separately.

Step 3: Differentiate the First Term

Differentiate \( aP^2 \). Using the power rule, the derivative \( \frac{d}{dP}(aP^2) = 2aP^{2-1} = 2aP \).

Step 4: Differentiate the Second Term

Differentiate \( bP^3 \). Using the power rule, the derivative \( \frac{d}{dP}(bP^3) = 3bP^{3-1} = 3bP^2 \).

Step 5: Combine the Derived Terms

Combine the derivatives of each term: \( \frac{dQ}{dP} = 2aP + 3bP^2 \). This represents the derivative of the function \( Q \) with respect to \( P \).

Key Concepts

Power Rule

The Power Rule is a fundamental concept in differentiation. It is a technique used to find the derivative of expressions like polynomials and other terms where variables are raised to a power. The Power Rule states that if you have a function in the form of a single term like \( f(P) = P^n \), then the derivative, \( f'(P) \), is equal to \( nP^{n-1} \). This simplifies the process of differentiating because you multiply the exponent by the coefficient and then reduce the exponent by one.

Consider a simple example with \( f(P) = P^5 \). Using the Power Rule, the derivative is \( f'(P) = 5P^{5-1} = 5P^4 \). You take 5, multiply it with the existing coefficient (which is 1 by default if not stated), and decrease the power by one. The beautiful part about the Power Rule is how swiftly it transforms differentiation into a mechanical operation of multiplying and subtracting, especially useful when dealing with polynomials.

• The power rule only works when the function is a power of a variable.
• Differentiate each term of a polynomial separately using this rule.

Polynomial Differentiation

Polynomial Differentiation is the process of finding the derivative of a polynomial function. Polynomials like \( Q = aP^2 + bP^3 \) are simply a sum of terms, each consisting of a coefficient and a variable raised to a power.

The process of differentiating a polynomial involves applying the Power Rule to each term in the polynomial separately. A polynomial in a single variable is just several power functions summed together, which makes it straightforward to apply the Power Rule iteratively across each term.

Take the polynomial \( Q = aP^2 + bP^3 \):

• The term \( aP^2 \) becomes \( 2aP \).
• Similarly, \( bP^3 \) differentiates to \( 3bP^2 \).

After differentiating each part, summing them back together gives the derivative of the polynomial. There’s a linearity in polynomial differentiation that makes it manageable and methodical, which is a major advantage when handling large polynomials with multiple terms. The key is consistency: ensure each part is differentiated before summing them up.

Derivative of Polynomials

The derivative of a polynomial is another polynomial. When you differentiate a polynomial function, the result is still a polynomial, usually of lower degree. This is due to the property of each term's power decreasing by one in the process.

The derivative tells us the rate of change of the polynomial at any given point. For the function \( Q = aP^2 + bP^3 \), its derivative \( \frac{dQ}{dP} = 2aP + 3bP^2 \) is a new polynomial derived from the original by applying the rules of differentiation.

• Each power in the polynomial is reduced by one degree after differentiating.
• The function’s behavior, such as increases and decreases, can be analyzed using its derivative.
• If the original polynomial is of degree \( n \), the derivative will be of degree \( n-1 \).

Polynomials are excellent for modeling because their derivatives provide clear insights into their changing behavior. Analyzing derivatives helps in understanding peaks, troughs, and points of inflection in the graph of the function.