The derivative of f(x) = exp(-x^2) is -2x * exp(-x^2).
To take the derivative of f(x) = exp(-x^2), you can use the power rule and chain rule. First, rewrite the function as f(x) = e^-x^2 and then apply the power rule to get f'(x) = -2x * e^-x^2. Finally, use the chain rule to replace e^-x^2 with its derivative -2x * e^-x^2.
The derivative of f(x) = exp(-x^2) is important because it is used to find the slope of the tangent line at any point on the curve. This is useful in many applications, such as optimization and physics problems.
The graph of f(x) = exp(-x^2) is a bell curve, also known as a Gaussian curve. Its derivative, -2x * exp(-x^2), is also a bell curve but with a steeper slope and a peak at x = 0. The derivative represents the slope of the tangent line at each point on the graph of f(x) = exp(-x^2).
Yes, there are many real-world applications of f(x) = exp(-x^2) and its derivative. Some examples include modeling the probability distribution of a random variable, calculating the electric field around a point charge, and describing the distribution of particles in a gas.