answer:**Analysis** This question mainly tests the expression of large numbers in scientific notation. The key is to master expressing a number greater than (10) in the form of (a×10^{n}), where (a) is a number with only one digit in its integer part, and (n) is a positive integer. Large numbers are generally expressed in scientific notation as (a×10^{n}), where the exponent (n=) the number of digits in the original number (-1), and (1 leqslant a < 10). **Solution** Solution: (51,800,000,000 = 5.18×10^{10}), Therefore, the answer is boxed{5.18×10^{10}}.

answer:Eleanor enjoyed meeting Marcus at the Athens conference and is interested in collaborating on a book project. The book would explore the impact of ancient cultural practices on modern society, and Eleanor believes Marcus's expertise would be invaluable. She suggests setting up a time to discuss the idea further.

answer:To find the derivative of e to the power of (x*x), we can use the chain rule. The chain rule states that the derivative of a composite function f(g(x)) is given by f'(g(x)) * g'(x). In this case, our composite function is e^(x*x), where the outer function f(u) = e^u and the inner function g(x) = x*x. The derivative of the outer function f(u) = e^u is simply f'(u) = e^u, since the exponential function is its own derivative. The derivative of the inner function g(x) = x*x is g'(x) = 2x, using the power rule of differentiation. Applying the chain rule, we get the derivative of e^(x*x) as f'(g(x)) * g'(x) = e^(x*x) * 2x = 2x * e^(x*x). Therefore, the derivative of e to the power of (x*x) is 2x * e^(x*x).

answer:To derive the formula for the derivative of cosine, we can use the definition of a derivative as a limit and the trigonometric identity for the cosine of a sum of angles. Recall the definition of the derivative of a function f(x) at a point x: f'(x) = lim(h → 0) [f(x + h) - f(x)]/h For the cosine function, we can write: (cos(x))' = lim(h → 0) [cos(x + h) - cos(x)]/h Using the cosine sum formula, we can rewrite cos(x + h) as: cos(x + h) = cos(x)cos(h) - sin(x)sin(h) Substituting this into our limit, we get: (cos(x))' = lim(h → 0) [cos(x)cos(h) - sin(x)sin(h) - cos(x)]/h We can simplify this expression by factoring out cos(x): (cos(x))' = lim(h → 0) [cos(x)(cos(h) - 1) - sin(x)sin(h)]/h Now, we can use two important limits: 1. lim(h → 0) (cos(h) - 1)/h = 0 2. lim(h → 0) sin(h)/h = 1 These limits can be proven using geometric arguments or the squeeze theorem. Applying these limits to our expression, we get: (cos(x))' = lim(h → 0) [-sin(x)sin(h)]/h = -sin(x) * lim(h → 0) sin(h)/h = -sin(x) * 1 = -sin(x) Therefore, the derivative of the cosine function is -sin(x).