Problem 15 - Calculus for Biology and Medicine

Question

In Problems 15-38, use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x}\left(1+t^{2}\right) d t $$

Step-by-Step Solution

Answer

The derivative \(\frac{dy}{dx} = 27x^2 + 3\).

Step 1: Understand Leibniz's Rule

Leibniz's Rule is a formula used to differentiate an integral whose limits are functions of the differentiation variable. It is given by: \(\frac{d}{dx}\int_{a(x)}^{b(x)}f(t)\,dt=f(b(x))b'(x)-f(a(x))a'(x)+\int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(t,x)\,dt\). In this problem, \(f(t) = 1 + t^2\), \(a(x) = 0\), and \(b(x) = 3x\).

Step 2: Apply Leibniz's Rule

Now, apply Leibniz's Rule to find the derivative. First, evaluate \(f(b(x))b'(x)\). Since \(b(x) = 3x\), we have \(b'(x) = 3\) and \(f(3x) = 1 + (3x)^2\). Thus, \(f(b(x))b'(x) = (1 + 9x^2) \cdot 3\). Next, we consider \(f(a(x))a'(x)\). Since \(a(x) = 0\), \(a'(x) = 0\), we have \(f(a(x))a'(x) = 0\).

Step 3: Calculate the Derivative

Since there is no explicit \(x\) in the function \(f(t) = 1 + t^2\), the partial derivative \(\frac{\partial}{\partial x}f(t,x)\) is zero. Thus, the integral part of Leibniz's Rule disappears. Therefore, the derivative is simply the expression from \(f(b(x))b'(x)\). Substitute: \((1 + 9x^2) \cdot 3 = 3 + 27x^2\). Therefore, \(\frac{dy}{dx} = 27x^2 + 3\).

Key Concepts

Understanding Calculus

Calculus is a branch of mathematics that studies continuous change. It has two major subfields: differential calculus and integral calculus. These fields provide tools for analyzing rates of change and accumulation of quantities.

Differential calculus focuses on the slope of curves and surfaces, essentially capturing how things change in the infinitesimally small scale. Meanwhile, integral calculus is concerned with the accumulation of quantities and the areas under or between curves.

Calculus is pivotal in many areas such as physics, biology, economics, and engineering. It allows us to understand complex systems by simplifying them into smaller, more manageable problems.

Integrals Simplified

An integral is a fundamental concept in calculus that signifies the accumulation of quantities. There are two main types of integrals: definite and indefinite integrals.

A definite integral is represented as \(\int_{a}^{b}f(x)\,dx\) where \(a\) and \(b\) are the bounds of integration. This integral computes the total accumulation of \(f(x)\) from \(a\) to \(b\). Indefinite integrals, on the other hand, are expressed as \(\int f(x)\,dx\), and they represent a family of functions whose derivative is \(f(x)\). These are expressed with a constant \(C\) to account for any potential horizontal shifts.

An integral can be thought of as finding the net area under a curve. In practical terms, consider that if \(f(x)\) represents speed, an integral can compute the distance traveled over a certain time period.

Deeper into Differentiation

Differentiation is the process of finding the derivative of a function, which measures how a function changes at any given point. The derivative of a function at a point is the slope of the tangent line at that point.

Derivatives have various practical applications. They can tell us about the speed of a moving object, optimize functions in economics to determine profit maximization or cost minimization, and even model real-world behavior in engineering systems.

Differentiation also forms the backbone of many advanced calculus concepts, such as Leibniz's rule used in the original problem. Leibniz's rule itself combines both differentiation and integration, allowing for the differentiation of integrals with variable limits. This is especially useful in dynamic systems where the boundaries can change uniformly with respect to another variable.