Problem 19 - Calculus for Biology and Medicine

Question

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{4}^{x^{2}+1} \sqrt{t} d t, x>0 $$

Step-by-Step Solution

Answer

The derivative \(\frac{d y}{d x}\) is \(2x \sqrt{x^2 + 1}\).

Step 1: Understanding Leibniz's Rule

Leibniz's rule is a technique used for differentiating under the integral sign. It is useful when you have an integral where the limits are functions of the variable you are differentiating with respect to. In this problem, the integral has an upper limit of integration that is a function of \(x\), \(x^2 + 1\).

Step 2: Apply Leibniz's Rule

According to Leibniz's rule: \(\frac{d}{dx} \int_{a(x)}^{b(x)} f(t, x)\, dt = f(b(x), x) \cdot \frac{d}{dx}b(x) - f(a(x), x) \cdot \frac{d}{dx}a(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}\, dt.\)However, in this case, \(a(x) = 4\) is a constant, so \(\frac{d}{dx}a(x) = 0\). Also, \(f(t) = \sqrt{t}\) does not explicitly depend on \(x\), so \(\frac{\partial f}{\partial x} = 0\). The expression simplifies to: \(\frac{d y}{d x} = \sqrt{b(x)} \cdot \frac{d}{dx}(b(x)).\)Here, \(b(x) = x^2 + 1\).

Step 3: Differentiate the Upper Limit

Differentiate the upper limit \(b(x) = x^2 + 1\) with respect to \(x\). This gives \(\frac{d}{dx}(x^2 + 1) = 2x\).

Step 4: Plug Values Into the Formula

Substitute \(b(x) = x^2 + 1\) and \(\frac{d}{dx}b(x) = 2x\) into the simplified Leibniz rule: \(\frac{d y}{d x} = \sqrt{x^2 + 1} \cdot 2x = 2x \sqrt{x^2 + 1}.\)So, the derivative of the integral with respect to \(x\) is \(2x\sqrt{x^2 + 1}\).

Key Concepts

Differentiation Under Integral Sign

Differentiation under the integral sign is a powerful technique in calculus that involves the differentiation of an integral whose limits or integrand depends on a variable. This concept shines when tackling problems involving integrals with variable, often non-trivial, limits that change as the input variable does.
One of the champions of this method is Leibniz's rule. This rule helps us systematically differentiate an integral by providing a clear framework:

• First, observe if the integrand depends explicitly on the variable.
• Second, check if the limits are functions of this variable.
• Then, apply Leibniz's formula to unpack the problem.

In our provided example, the integrand √t is independent of the variable x. However, the upper limit is not a constant—it changes as f(x) = x² + 1 does. Hence, an application of Leibniz's rule is essential to differentiate this type of integral effectively.

Variable Limits of Integration

In calculus, dealing with variable limits of integration adds another layer of complexity to finding derivatives of integrals. These limits change their values as the variable of differentiation changes, often making it necessary to adjust our approach.
In Leibniz's rule, we account for this variability by differentiating the limit itself and assessing how this change affects the integral. The formula suggests examining:

• The direct contribution of the variable limit to the value of the integral.
• How the function evaluated at these variable limits changes as the variable changes.

For our specific integral, the upper limit x² + 1 introduces a dependency on x. Differentiating this limit, specifically \( \rac{d}{dx}(x^2 + 1) = 2x \), is crucial for accurately applying Leibniz's formula. We thus obtain the integral's derivative by further multiplying by \( \sqrt{x^2 + 1} \). This scenario reflects just how important it is to adjust our calculus processes to accommodate variable limits for integration.

Calculus in Biology

Calculus finds its way into biology in numerous ways, often via processes that can be modeled as integrals or derivatives. From population models to individual organism growth rates, calculus provides the foundational tools for describing dynamic biological systems.
Differentiation under the integral sign, especially when dealing with variable limits, can be particularly useful. Consider modeling the spread of a substance across a biological membrane, where the rate of flow can depend on varying concentrations over time—an integral setup with variable limits. Or the rate of enzyme activity might vary with substrate concentration, where you again use integrals to describe overall activity across a range of conditions.

• Modeling dynamic changes across cell membranes.
• Description of growth rates where conditions change over time.

These examples highlight differentiated approaches to such integrals, and how insightful it can be when the limits change with respect to a biological variable like time or concentration, giving you a precise mathematical framework to analyze biological systems.