Problem 53 - University Calculus: Early Transcendentals

Question

Find \(d y / d x\). $$y=\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}} d t$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 2x e^{x^2 - x} \).

Step 1: Identify the Structure of the Problem

The problem requires finding the derivative of an integral with respect to its upper limit, involving the function in the upper limit of integration. This is a good candidate for using the Fundamental Theorem of Calculus combined with the Chain Rule.

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if you have an integral of the form \(y = \int_{a}^{u(x)} f(t) \, dt\), the derivative of \(y\) with respect to \(x\) is \(\frac{d}{dx} \left( \int_{a}^{u(x)} f(t) \, dt \right) = f(u(x)) \cdot \frac{d}{dx} [u(x)]\).

Step 3: Determine the Values

Given \(y = \int_{0}^{e^{x^2}} \frac{1}{\sqrt{t}} \, dt\), here \(f(t) = \frac{1}{\sqrt{t}}\) and \(u(x) = e^{x^2}\). Substitute these into the formula derived from the theorem: \(\frac{d}{dx}\left( \int_{0}^{e^{x^2}} \frac{1}{\sqrt{t}} \, dt \right) = \frac{1}{\sqrt{e^{x^2}}} \cdot \frac{d}{dx} [e^{x^2}]\).

Step 4: Calculate the Respective Derivatives

First, calculate \(\frac{1}{\sqrt{e^{x^2}}}\), which simplifies to \(e^{-x}\). Then, find the derivative of the upper limit, \(\frac{d}{dx} [e^{x^2}] = e^{x^2} \cdot 2x\).

Step 5: Combine the Results

Combine these results together: \( \frac{dy}{dx} = e^{-x} \, \cdot e^{x^2} \, \cdot 2x = 2x e^{x^2 - x}\).

Key Concepts

Chain Rule

The Chain Rule is a key concept in calculus used to differentiate composite functions. It's particularly handy when dealing with functions nested inside other functions. Imagine a situation where you have a function written as \( h(x) = f(g(x)) \). If you want to differentiate \( h(x) \) with respect to \( x \), the Chain Rule provides an efficient method: take the derivative of the outer function \( f \) evaluated at the inner function \( g(x) \), then multiply it by the derivative of the inner function \( g \).

• Formula: \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).
• In our exercise, the function \( u(x) = e^{x^2} \) is the inside function, while \( \frac{1}{\sqrt{t}} \) is the function being integrated.
• Apply this by first differentiating \( e^{x^2} \) to get \( 2x\cdot e^{x^2} \) and then multiplying by \( f(u(x)) \), which in this case is \( \frac{1}{\sqrt{e^{x^2}}} \).

The Chain Rule is crucial when functions inside another function change, such as upper limits in definite integrals depending on \( x \). It simplifies seemingly complex derivatives into manageable calculations.

definite integral

A definite integral calculates the area under a curve between two specific points. In notation, this is often written as \( \int_{a}^{b} f(t) \, dt \), where \( a \) and \( b \) are the lower and upper limits of integration, respectively. This integral gives a single numerical result.

• The process involves finding the antiderivative (indefinite integral) first, then calculating at the bounds \( b \) and \( a \), and finally subtracting the two values.
• In our specific exercise, the integral \( \int_{0}^{e^{x^2}} \frac{1}{\sqrt{t}} \, dt \) computes the area from 0 to \( e^{x^2} \) under the function \( \frac{1}{\sqrt{t}} \).
• The presence of an upper limit that depends on \( x \) links back to the Chain Rule and the Fundamental Theorem of Calculus.

Definite integrals serve as the backbone for various applications in physics, engineering, and probability, providing solutions to problems involving rate of change and total accumulation.

upper limit of integration

The upper limit of integration plays a significant role in defining the scope of the area being considered under a curve in a definite integral. When this limit is a function of \( x \), it introduces an interesting dimension to solving calculus problems.

• If the upper limit is a constant, the definite integral provides the fixed area between the curve and the x-axis from \( a \) to \( b \).
• When the upper limit is variable as in \( \int_{0}^{e^{x^2}} \frac{1}{\sqrt{t}} \, dt \), it requires a tactical approach to differentiation, invoking both the Fundamental Theorem of Calculus and the Chain Rule.
• This problem-solving method helps find the rate of change of the integral's area concerning the changes in the upper limit (i.e., \( e^{x^2} \) for our exercise).

Handling variable upper limits is essential for understanding dynamic systems, ensuring accurate modeling of varying quantities in everyday contexts, such as economics or natural sciences.