Problem 33 - Calculus for Biology and Medicine

Question

Differentiate $$ g(t)=a^{3} t-a t^{3} $$

Step-by-Step Solution

Answer

The derivative of the function is \( g'(t) = a^3 - 3at^2 \).

Step 1: Identify the Function Components

The function given is \( g(t) = a^3 t - a t^3 \), where \(a\) is a constant and \(t\) is the variable. We need to find the derivative of this function with respect to \(t\).

Step 2: Apply the Power Rule and Constant Rule

Recall that the derivative of \(c \cdot t^n\) where \(c\) is a constant, is \(n \cdot c \cdot t^{n-1}\). For the first term \(a^3 t\), treat \(a^3\) as a constant multiplied by \(t\). Its derivative is \(a^3\cdot 1 = a^3\).

Step 3: Differentiate the Second Term

For the second term \(-a t^3\), using the power rule, differentiate to get: \(-3 a t^{2}\).

Step 4: Combine the Derivatives

Add the results of the derivatives of both terms to get the complete derivative of the function. Thus, the derivative is: \[ g'(t) = a^3 - 3at^2 \].

Key Concepts

The Power Rule

The power rule is a fundamental rule in calculus used to differentiate functions of the form \(x^n\), where \(x\) is a variable and \(n\) is a real number. This rule simplifies the process by providing a straightforward formula for finding derivatives.
When you have a function \(f(x) = x^n\), the derivative, denoted as \(f'(x)\), is calculated as \(n \cdot x^{n-1}\). Essentially, you multiply the exponent \(n\) by the variable base, and then subtract one from the exponent.
For example, if you need to differentiate \(x^3\), you use the power rule: multiply 3 by \(x\) and reduce the power by one, giving \(3x^2\).
This process is handy for quickly differentiating polynomials, which often consist of terms like \(x^2\), \(x^3\), and so forth. Each term can be differentiated independently using the power rule, making it an essential tool in calculus.

The Constant Rule

The constant rule in differentiation is one of the simplest rules to apply. It states that the derivative of a constant is zero. This is because a constant value does not change, and therefore, its rate of change (or slope) is zero.
In mathematical terms, if \(c\) is a constant, then the derivative \(\frac{d}{dx}[c] = 0\).
Additionally, if you have a term that is a constant multiplied by a variable raised to a power, such as \(c \cdot x^n\), the constant rule works in tandem with the power rule. Here, the constant \(c\) remains as a coefficient when you differentiate.
For instance, when differentiating \(3x^2\), treat 3 as a constant. Using the power rule, you have \(2 \cdot 3 \cdot x^{2-1} = 6x\). The constant \(3\) merely multiplies the result of the derivative of \(x^2\). This way, the constant rule simplifies handling terms that include constants.

Polynomial Differentiation

Polynomial differentiation involves finding the derivative of polynomial expressions, which are sums of multiple terms where each term is a variable raised to a power times a constant coefficient. The key to differentiating polynomials efficiently lies in applying the power rule and constant rule across each term.
Given a polynomial function like \(P(x) = c_n x^n + c_{n-1} x^{n-1} + \ldots + c_1 x + c_0\), you differentiate each term individually because they'll often be in the form that the power and constant rules apply to.
For example, take the polynomial \(g(t) = a^3 t - a t^3\):

• The first term, \(a^3 t\), is differentiated to \(a^3\) because \(a^3\) is a constant and the power of \(t\) is 1. The derivative of \(t^1\) is simply 1, leaving just \(a^3\).
• The second term, \(-a t^3\), uses the power rule. You multiply \(-3\) by \(-a t^2\), yielding \(-3at^2\).

Combining these results, the derivative of the entire polynomial \(g(t)\) becomes \(g'(t) = a^3 - 3at^2\). This step-by-step application of rules makes polynomial differentiation systematic and manageable.