The exponential function f(x)=e^x can be defined for...

The exponential function $f(x)=e^{x}$ can be defined for exponents that are complex numbers, $a+b i,$ where $i=\sqrt{-1}$. Complex-valued functions may be differentiated just like real-valued functions, and previous derivative rules carry over in this setting. For example, $\left(e^{a x}\right)^{\prime}=a e^{a x}$ holds if $a$ and $x$ are complex numbers. Furthermore, in the complex-number domain there is an important formula (known as Euler's formula) relating exponential and trigonometric functions: $e^{i x}=\cos x+i \sin x$. (These topics are all addressed formally when we develop the theory of power series in Chapter $10 .)$ (a) Using Euler's formula, prove $$ \sin x=\frac{e^{i x}-e^{-i x}}{2 i} \text { and } \cos x=\frac{e^{i x}+e^{-i x}}{2} $$ (b) Using the expressions from (a) and the derivative rule for $e^{x}$, prove the derivative rules for $\sin x$ and $\cos x$.

The exponential function $f(x)=e^{x}$ can be defined for exponents that are complex numbers, $a+b i,$ where $i=\sqrt{-1}$. Complex-valued functions may be differentiated just like real-valued functions, and previous derivative rules carry over in this setting. For example, $\left(e^{a x}\right)^{\prime}=a e^{a x}$ holds if $a$ and $x$ are complex numbers. Furthermore, in the complex-number domain there is an important formula (known as Euler's formula) relating exponential and trigonometric functions: $e^{i x}=\cos x+i \sin x$. (These topics are all addressed formally when we develop the theory of power series in Chapter $10 .)$
(a) Using Euler's formula, prove
$$
\sin x=\frac{e^{i x}-e^{-i x}}{2 i} \text { and } \cos x=\frac{e^{i x}+e^{-i x}}{2}
$$
(b) Using the expressions from (a) and the derivative rule for $e^{x}$, prove the derivative rules for $\sin x$ and $\cos x$.

Key Concepts

Differentiation of Complex Exponential Functions

The rules of differentiation that apply to real-valued functions extend naturally to complex-valued functions. In particular, the derivative of the complex exponential function e^(ax) with respect to x is a e^(ax), where a can be a complex number. This property is essential when differentiating the exponential forms of sine and cosine functions derived via Euler's formula.

Euler's Formula

Euler's Formula states that for any real number x, the complex exponential function satisfies the relation e^(ix) = cos x + i sin x. This fundamental formula provides a deep connection between exponential functions with imaginary exponents and trigonometric functions, underpinning many results in complex analysis and engineering.

Exponential Representation of Trigonometric Functions

Using Euler's formula and its counterpart for negative exponents, trigonometric functions such as sine and cosine can be expressed as combinations of complex exponentials. This representation simplifies many proofs and computations by converting trigonometric identities into exponential form, as seen in the formulas sin x = (e^(ix) - e^(-ix))/(2i) and cos x = (e^(ix) + e^(-ix))/2.

Derivative Rules for Trigonometric Functions

By expressing sine and cosine in their exponential forms and applying the standard differentiation rules for exponential functions, one can derive the derivative formulas for trigonometric functions. This approach illustrates how the complex representation streamlines the process of differentiation and serves as a bridge between exponential and trigonometric functions in calculus.

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