Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x^{2}} e^{-t^{2}} d t $$

Step-by-Step Solution

Answer

\( G'(x) = 2x \cdot e^{-x^4} \)

Step 1: Recognize the Formula

Recognize that the given function is in the form of Leibniz's Rule, which states that if \( F(x) = \int_{a(x)}^{b(x)} f(t) \, dt \), then \( F'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \).

Step 2: Identify the Bounds

In the given problem, the bounds are \( a = 1 \) and \( b(x) = x^2 \). Thus, \( a'(x) = 0 \) and \( b'(x) = 2x \).

Step 3: Apply Leibniz Rule

Since \( a = 1 \) is a constant, its derivative is zero, thus the expression simplifies to \[ G'(x) = e^{-(x^2)^2} \cdot 2x - 0 = 2x \cdot e^{-x^4} \].

Step 4: Final Derivative

Combine the elements from the previous steps to give the derivative of the given function. Therefore, \( G'(x) = 2x \cdot e^{-x^4} \).

Key Concepts

Understanding Derivative Calculation

The process of finding the derivative of a function is called derivative calculation. It helps in determining how a function changes as its input changes. In our exercise, we use the Leibniz Rule to find the derivative of a function given as an integral. A derivative represents the slope or rate of change at any given point of the original function. When working with integrals, like in our problem, the goal is to find how the integral's value changes when its upper or lower limits are functions of another variable, such as \( x \). The derivative calculation involves:

• Identifying the structure of the function in terms of standard forms, like the integral in Leibniz's Rule.
• Using known rules, such as identifying the changing bounds and their derivative when plugged into the integral's function.
• Paying close attention to how the derivative of the bounds (here \( b'(x) \)) affects the overall calculation.

This systematic approach helps in simplifying tasks of determining how complicated functions, like integrals, behave as their input changes.

Exploring Integration Bounds

Integration bounds in calculus define the limits or boundaries over which an integral operates. In our integral problem, these bounds are from \(1\) to \(x^2\).

• The lower bound \(a = 1\) is constant, meaning that it does not change as \(x\) changes, hence its derivative \(a'(x)\) is zero.
• The upper bound \(b(x) = x^2\) is a function of \(x\), meaning it changes with \(x\). The derivative \(b'(x) = 2x\) is crucial as it shows how fast the upper bound is changing as \(x\) changes.

When dealing with bounds in integral calculus, especially for variable limits, it is important to compute each limit's derivative correctly. This derivative directly influences the overall application of definitions like the Leibniz Rule in determining the final outcome of problems involving integrals.

Effective Calculus Problem-Solving Strategies

Solving calculus problems requires a range of strategic approaches. Problems involving derivatives and integrals, like our given exercise, benefit from a structured method:

• Understanding the core concept: Begin by clearly identifying which calculus principles apply, such as Leibniz Rule, chain rule, or fundamental theorem(s) of calculus.
• Systematic breakdown: Tackle each part of the problem separately. Identify all components, like the function being derived, and the bounds if integrals are involved.
• Leveraging known rules: Apply appropriate rules systematically, as done with Leibniz's Rule, analyzing each limit and assessing their influences through derivatives.
• Consistent review: Always recheck your steps and computations to ensure accuracy, just as when deriving \( G'(x) \) to verify each component is correctly included in the final expression.

These strategies will not only enhance your problem-solving skills but also build a strong intuition for tackling complex tasks in calculus.