91Ó°ÊÓ

Differentiation is a core concept in calculus, and it focuses on finding the rate at which a function changes at any given point. In simpler terms, differentiation allows us to determine how the output of a function changes as the input changes.

For a function of a variable like \( y = f(x) \), its derivative \( \frac{dy}{dx} \) represents the slope of the tangent line to the curve at any point \( x \). This slope gives the rate of change of the function with respect to \( x \).

• First Principle: Differentiation can be understood using the limit definition. For a small change \( \Delta x \), the change in \( y \) is \( \Delta y = f(x+\Delta x) - f(x) \). The derivative is obtained by taking the limit as \( \Delta x \) approaches zero:\[ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \]
• Product Rule: When differentiating a product of two functions, \( u(x)v(x) \), use the rule:
  \[ (uv)\,' = u'v + uv' \]

In our exercise, we directly used the Fundamental Theorem of Calculus which simplifies the differentiation process for an integral function.

Integral calculus is the branch of calculus dealing with integrals and their properties. It involves accumulating quantities, such as areas under curves or total change given a rate of change. There are two main types of integrals: indefinite and definite.

• Indefinite Integrals: These represent families of functions and contain an arbitrary constant, \( C \). They describe antiderivatives, or functions whose derivative returns the original function.
• Definite Integrals: These calculate the net area under a curve from one point to another, providing a specific numerical value. The Fundamental Theorem of Calculus ties them to differentiation.

In our exercise, we dealt with a definite integral, \( \int_{3}^{x} u e^{4u} \, du \), representing a function that accumulates values as \( u \) changes from 3 to \( x \). The theorem shows us that the derivative of this integral with respect to \( x \) is simply the original integrand evaluated at \( x \): \( x e^{4x} \). This result highlights how differentiation and integration are fundamentally linked processes in calculus.

Functions of a variable are mathematical expressions that describe relationships between quantities. A function assigns an output to every input in its domain. For instance, in \( y = f(x) \), \( x \) is the independent variable, and \( y \) is the dependent variable.

The concept of a function is essential in calculus because it allows us to study how a dependent variable changes as an independent variable varies.

• Domain and Range: The domain is the set of all possible inputs (\( x \) values), whereas the range is the set of all potential outputs (\( y \) values).
• Continuous Functions: A function is continuous if there are no breaks, holes, or jumps in its graph. Many calculus techniques assume functions are continuous over their domain.

In the given exercise, the function \( y = \int_{3}^{x} u e^{4u} \, du \) is defined in terms of an integral, emphasizing the relationship between the variable \( x \) and the output \( y \). Understanding how to differentiate this function shows how calculus helps us navigate complex relationships between variables.