Final answer:
The derivative of sin(x) using exponential notation involves the complex exponential function e^(ix) and Euler's formula. The derivative of e^(ix) with respect to x is ie^(ix), which is equal to cos(x) + isin(x). Hence, the derivative of sin(x) is icos(x). The derivative of cos(x) + isin(x) using Euler's formula is icos(x) - sin(x). The derivative of e^(-iπ/2) is 0. Finally, the derivative of e^(iθ) with respect to x at θ = x is ie^(ix).
Step-by-step explanation:
The derivative of sin(x) using exponential notation can be found by considering the complex exponential function e^(ix) and Euler's formula e^(ix) = cos(x) + isin(x) .
a) The derivative of e^(ix) with respect to x can be found using the chain rule. Since the derivative of e^(ix) is ie^(ix), the derivative of sin(x) is icos(x).
b) Using Euler's formula, we can express cos(x) + isin(x) as e^(ix). Taking the derivative of e^(ix) with respect to x gives us ie^(ix), which is equal to icos(x) + isin(x). Therefore, the derivative of cos(x) + isin(x) is icos(x) - sin(x).
c) The derivative of e^(-iπ/2) does not involve the variable x, so its derivative is 0.
d) To find the derivative of e^(iθ) with respect to x at θ = x, we can differentiate with respect to θ first, giving us ie^(iθ). Then, we substitute θ = x to obtain the derivative of e^(iθ) with respect to x at θ = x, which is ie^(ix).