A derivative is a mathematical concept that measures the rate of change of a function at a specific point. It tells us how the output of a function changes when the input changes.
To find the derivative of a function, we use the rules of differentiation. These include the power rule, product rule, quotient rule, and chain rule. We also need to know the basic derivatives of common functions such as polynomials, exponentials, and trigonometric functions.
Finding the minimum and maximum of a function helps us understand the behavior of the function and identify important points, such as the highest or lowest points, on the graph. It can also help us optimize a function to find the best solution to a problem.
To find the minimum and maximum of a function, we first take the derivative of the function and set it equal to 0. Then, we solve for the variable to find the critical points. We can then plug these critical points back into the original function to determine the minimum and maximum values.
Sure! First, we take the derivative of the function using the power rule and chain rule. This gives us the equation -e^(-x) + 2e^(-2x). Next, we set this equal to 0 and solve for x, which gives us x = ln(2). Plugging this value back into the original function, we get a minimum value of -1/2 at x = ln(2) and a maximum value of -1 at x = 0.