Problem 9 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable.$$ h(t)=-\frac{1}{3} t^{4}+4 t $$

Step-by-Step Solution

Answer

The derivative of \( h(t) = -\frac{1}{3} t^4 + 4t \) is \( -\frac{4}{3} t^3 + 4 \).

Step 1: Identify Components

The function given is \( h(t) = -\frac{1}{3} t^4 + 4t \). It is composed of two terms: \(-\frac{1}{3} t^4\) and \(4t\). We need to differentiate each term with respect to the variable \( t \).

Step 2: Differentiate \( -\frac{1}{3} t^4 \)

Use the power rule to differentiate \( -\frac{1}{3} t^4 \). The power rule states that \( \frac{d}{dt} (t^n) = n \cdot t^{n-1} \). Applying this, we get \( \frac{d}{dt} (-\frac{1}{3} t^4) = -\frac{1}{3} \cdot 4 \cdot t^{4-1} = -\frac{4}{3} t^3 \).

Step 3: Differentiate \( 4t \)

Differentiating \( 4t \) with respect to \( t \) using the basic derivative rule \( \frac{d}{dt} (kt) = k \) gives \( \frac{d}{dt} (4t) = 4 \).

Step 4: Combine the Results

Combine the derivatives of both terms: \( \frac{d}{dt} h(t) = -\frac{4}{3} t^3 + 4 \). This is the derivative of the function \( h(t) \).

Key Concepts

Power Rule

The power rule is a favorite among calculus students for its simplicity and effectiveness. It helps us find derivatives of expressions when they are in the form of a power function. A typical power function looks like this: \( f(x) = x^n \). According to the power rule, the derivative of this function is given by \( f'(x) = n \cdot x^{n-1} \).
To illustrate, think of \( h(t) = -\frac{1}{3} t^4 \). Applying the power rule, we see that the exponent 4 comes down in front as a coefficient, and we decrease the exponent by one, resulting in the derivative \( -\frac{4}{3} t^3 \). This technique is essential to differentiate polynomials and makes calculating derivatives quick and straightforward.
It’s important to note that the power rule only applies to terms where the variable is raised to a constant power. If you apply it effectively, it will save you time and help you tackle more complex calculus problems with ease.

Derivative

Understanding derivatives is like unlocking a new way to see the world of functions. At its core, a derivative represents the rate of change of a function with respect to one of its variables. For example, when we talk about the derivative of a function \( h(t) \), we're interested in how \( h(t) \) changes as \( t \) changes.
This rate of change tells us how fast a function's value is increasing or decreasing, thereby offering a glimpse into the function's behavior across its domain. In practical terms, if a car's position over time is expressed as a function, the derivative of that function will tell you the car's velocity.
In practice, deriving a function will often yield another function that describes how the original function behaves in terms of its slope. In our given problem, after deriving \( h(t) = -\frac{1}{3} t^4 + 4t \), we obtain \( -\frac{4}{3} t^3 + 4 \), a new function that precisely describes how \( h(t) \) changes as \( t \) changes. Recognizing the significance of derivatives can transform how you apply calculus to real-world issues.

Calculus

Calculus is often described as the mathematics of change. It's an area of mathematics that explores how things change and develops concepts to predict behavior and outcomes of these changes. There are two main branches: differential calculus, which deals with derivatives, and integral calculus, which involves integrals.
Differentiation, as explored in this exercise, is a key component of calculus. It provides the tools to determine the instantaneous rate of change or the slope of a curve at any given point. This is crucial across different fields like physics, engineering, economics, and even biology.
Moreover, calculus is not just an academic endeavor. Whether you are solving problems related to motion, growth, optimization, or even analyzing statistical data, calculus serves as a foundational pillar. Grasping its concepts, like the power rule and derivatives, empowers students to tackle more advanced topics and gain insights into various phenomena in both natural and social sciences.