Question

Prove that the derivative has the following properties: $$ \frac{d}{d t}\left(\varphi_{1}+\varphi_{2}\right)=\frac{d \varphi_{1}}{d t}+\frac{d \varphi_{2}}{d t}, \quad \frac{d}{d t}\left(\varphi_{1} \varphi_{2}\right)=\frac{d \varphi_{1}}{d t} \varphi_{2}+\varphi_{1} \frac{d \varphi_{2}}{d t} $$ In particular the derivative of a polynomial with complex coefficients is given by the same formula as in the case of real coefficients. If \(n>1\), it is impossible to multiply two curves with values in \(C^{n}\), How ever, since \(C^{n}\) is a \(C\)-module, it is possible to multiply a curve \(\varphi: I \rightarrow C^{n}\) by a function \(f: I \rightarrow C\) $$ (f \varphi)(t)=f(t) \varphi(t) $$

Short answer

Answer: The derivative of a polynomial with complex coefficients is obtained by applying the same formula as in the case of real coefficients, which is: $$ \frac{dp}{dt} = 0 + c_1 + 2c_2t + \cdots + nc_nt^{n-1} $$ where \(c_i\) are the complex coefficients of the polynomial.

Step-by-step solution

Prove the first property of the derivative

To prove the first property, we need to show that the derivative of the sum of two differentiable functions \(\varphi_1(t)\) and \(\varphi_2(t)\) is equal to the sum of their derivatives. Let us consider the sum of the functions: $$ (\varphi_1 + \varphi_2)(t) = \varphi_1(t) + \varphi_2(t) $$ Now, let us differentiate this sum with respect to the variable \(t\): $$ \frac{d}{dt}(\varphi_1 + \varphi_2) = \frac{d}{dt}(\varphi_1(t) + \varphi_2(t)) $$ By the linearity of the derivative, we can simply sum the individual derivatives of the two functions: $$ \frac{d}{dt}(\varphi_1(t) + \varphi_2(t)) = \frac{d\varphi_1}{dt} + \frac{d\varphi_2}{dt} $$ Hence, the first property has been proven.

Prove the second property of the derivative

To prove the second property, we should consider the product of two differentiable functions \(\varphi_1(t)\) and \(\varphi_2(t)\): $$ (\varphi_1\varphi_2)(t) = \varphi_1(t) \varphi_2(t) $$ Now, apply the product rule for differentiation: $$ \frac{d}{dt}(\varphi_1 \varphi_2) = \frac{d\varphi_1}{dt} \varphi_2(t) + \varphi_1(t) \frac{d\varphi_2}{dt} $$ The second property has now been proven as well.

Derivative of a polynomial with complex coefficients

Let us consider a polynomial with complex coefficients, which is defined as follows: $$ p(t) = c_0 + c_1t + c_2t^2 + \cdots + c_nt^n, \quad c_i \in \mathbb{C} $$ To show that the derivative of this polynomial follows the same formula as in the case of real coefficients, we must differentiate this polynomial with respect to \(t\) and use the properties previously proven: $$ \frac{dp}{dt} = \frac{d}{dt}(c_0) + \frac{d}{dt}(c_1t) + \frac{d}{dt}(c_2t^2) + \cdots + \frac{d}{dt}(c_nt^n) $$ Now, we can determine the derivative of each term using the linearity property: $$ \frac{dp}{dt} = 0 + c_1 + 2c_2t + \cdots + nc_nt^{n-1} $$ This formula demonstrates that the derivative of a polynomial with complex coefficients follows the same rule as in the case of real coefficients.

Key concepts

Derivative properties

The concept of derivative properties is fundamental in calculus. It refers to how derivatives behave under different operations. One key property is the linearity of derivatives. This means that the derivative of a sum is simply the sum of the derivatives. For example, if we have two differentiable functions, say \( \varphi_1(t) \) and \( \varphi_2(t) \), the derivative of their sum would be: \[\frac{d}{dt}(\varphi_1 + \varphi_2) = \frac{d \varphi_1}{dt} + \frac{d \varphi_2}{dt}\]This property shows how addition in function space translates directly into addition in derivative space.
Another important property is related to multiplication, which is somewhat more complex due to the interaction between the changing rates of the functions involved.
By understanding these derivative properties, students can appreciate how calculus simplifies the manipulation of complex mathematical expressions.

Complex polynomials

Complex polynomials are an intriguing extension of real polynomials, yet they follow similar differentiation rules. A complex polynomial is a function like \( p(t) = c_0 + c_1t + c_2t^2 + \cdots + c_nt^n \), where each coefficient \( c_i \) is a complex number. The derivative of a polynomial looks at how the output changes as the input varies, respecting the coefficients' magnitude and direction.
The great news is, differentiating complex polynomials uses the same principles as real ones. By applying familiar rules for derivatives, we can determine the derivative of each term. For instance:

• The derivative of \( c_1t \) is \( c_1 \)
• The derivative of \( c_2t^2 \) is \( 2c_2t \)
• And so forth, ending with \( nc_nt^{n-1} \)

So, the derivative of \( p(t) \) becomes:
\[\frac{dp}{dt} = 0 + c_1 + 2c_2t + \cdots + nc_nt^{n-1}\]
This consistency across real and complex domains helps simplify calculations and deepen our understanding of polynomial behavior.

Product rule

The product rule is a handy tool in calculus that allows us to differentiate products of two functions. When dealing with two functions, \( \varphi_1(t) \) and \( \varphi_2(t) \), their product demands more than merely multiplying their individual derivatives.
The product rule states:

• First, differentiate the first function and multiply it by the second function.
• Then, differentiate the second function and multiply it by the first function.
• Finally, sum these two results.

Mathematically, this is expressed as:
\[\frac{d}{dt}(\varphi_1 \varphi_2) = \frac{d \varphi_1}{dt} \cdot \varphi_2 + \varphi_1 \cdot \frac{d \varphi_2}{dt}\]
This rule ensures that we account for how each function changes while considering their individual rates of change. The product rule is essential when working with more complex expressions, aiding in accurately finding their total rate of change.