Problem 16 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Bestimmen Sie jeweils aie Ableitung der folgenden differenzierbaren Funktionen a) \(f: R \rightarrow\) mit \(f(t):=t(1-i t) .\)

Step-by-Step Solution

Answer

The derivative is \( f'(t) = 1 - 2it \).

Step 1: Identify the function and the variables

The function given is \( f(t) = t(1 - it) \). Here, \( t \) is the variable, and \( i \) is the imaginary unit where \( i^2 = -1 \). We aim to find the derivative of this function with respect to \( t \).

Step 2: Simplify the function

Simplify the function by applying the distributive property. The function can be rewritten as \( f(t) = t - it^2 \).

Step 3: Differentiate each term

Differentiate each term of the function separately. The derivative of \( t \) with respect to \( t \) is 1. The derivative of \( -it^2 \) with respect to \( t \) is \(-2it\).

Step 4: Combine the derivatives

Combine the derivatives obtained in the previous step: the derivative of \( f(t) \) is \( 1 - 2it \). Thus, the derivative of the function \( f(t) = t(1 - it) \) is \( f'(t) = 1 - 2it \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It deals with finding the rate at which a function changes with respect to a variable. In simpler terms, it tells us how the output of a function changes as the input changes.

When we differentiate a function, we are essentially finding its derivative. This is crucial for understanding the behavior of functions in various fields like physics, engineering, and economics. Differentiation helps in determining slopes of curves, rates of change, and can even find the maxima and minima of functions.

In the given exercise, we focused on differentiating a function of a real variable. The basic rules of differentiation, such as the power rule, product rule, and chain rule, were applied to find the derivative. Remember, practice is key to mastering differentiation.

Complex functions

Complex functions involve numbers that have a real and an imaginary component. An imaginary unit is represented by \( i \), where \( i^2 = -1 \).

Working with complex functions can initially seem intimidating, but they follow similar principles to regular functions. When dealing with complex functions, you separate them into their real and imaginary components. This separation helps simplify and manage calculations.

In our example, the function \( f(t) = t(1 - it) \) involves a complex part due to \( i \), making both its real and imaginary parts essential. Simplifying the function helps perform operations smoothly, leading us to effectively differentiate complex functions.

Derivative rules

Derivative rules are the toolset used to compute the derivative of different types of functions. They provide shortcuts and methods, bypassing the need for complicated limit processes.

Key rules include:

• Power rule: The derivative of \( t^n \) is \( nt^{n-1} \).
• Product rule: Used when differentiating the product of two functions, \( (u \, v)' = u' \, v + u \, v' \).
• Chain rule: For a composite function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

In the solution, the simplification phase allows direct application of these rules. The disentangled terms make it straightforward to use the power rule on each term of the function individually, producing the solution \( 1 - 2it \).