Final Answer:
1. d/dx*(-e^-x) = e^-x
2. d/dx*(e^-x) = -e^-x
3. d/dx*(-1·e^-x) = -e^-x
4. d/dx*(1·e^-x) = e^-x
Step-by-step explanation:
When finding the derivative of a function, we are essentially finding the slope of the tangent line at a specific point on the curve. In other words, we are determining how the function changes with respect to the independent variable, in this case, x. To find the derivatives of the given expressions, we will use the chain rule, which states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
1. d/dx*(-e^-x):
We can rewrite this expression as -1*e^-x. The outer function is -1, and the inner function is e^-x. Applying the chain rule, we get the derivative of the outer function, which is 0, multiplied by the derivative of the inner function, which is e^-x. Therefore, the final answer is e^-x.
2. d/dx*(e^-x):
Here, the outer function is 1, and the inner function is e^-x. Following the same steps as above, we get the derivative of the outer function, which is 0, multiplied by the derivative of the inner function, which is e^-x. However, since the outer function is 1, the final answer for this expression is -e^-x.
3. d/dx*(-1·e^-x):
For this expression, we have both an outer function (-1) and an inner function (e^-x). Applying the chain rule, we get the derivative of the outer function, which is -1, multiplied by the derivative of the inner function, which is e^-x. Therefore, the final answer is -e^-x.
4. d/dx*(1·e^-x):
This expression is similar to the previous one, but the outer function is now 1 instead of -1. Following the same steps, we get the derivative of the outer function, which is 1, multiplied by the derivative of the inner function, which is e^-x. Thus, the final answer for this expression is e^-x.
In conclusion, the derivatives of the given expressions are e^-x, -e^-x, -e^-x, and e^-x respectively. These derivatives represent the instantaneous rate of change of the original functions with respect to x. By using the chain rule, we were able to find the derivatives efficiently and accurately. It is important to note that for all exponential functions, the derivative is the same as the original function, with the exception of a constant coefficient.