JasonX said:please explain in more detail on how we come up with the answers below. Thanks in advance!
(formulas much appreciated)
Differentiate:
- 1
y=e^x
=e^x
- 2
y=lnx
=1/x
The process for differentiating y=e^x is known as the power rule. This rule states that the derivative of e^x is simply e^x itself. In other words, the derivative of y=e^x is equal to e^x.
Sure, let's take y=e^x as an example. The derivative of y=e^x is simply e^x. So, dy/dx = e^x. This means that the slope of the graph of y=e^x at any point is equal to the value of e^x at that point.
If the function involves e^x raised to a power, such as y=e^(2x), the power rule still applies. In this case, the derivative would be 2e^(2x).
Yes, there are other rules for differentiating e^x. One important rule is the chain rule, which is used when differentiating composite functions involving e^x. There is also the product and quotient rule for differentiating functions that involve e^x multiplied or divided by other functions.
Differentiating y=e^x is useful in a variety of scientific fields, including physics, chemistry, and biology. For example, in physics, the rate of change of an exponential function can be found by differentiating it. In chemistry, the rate of a chemical reaction can be determined by differentiating the concentration function, which often involves e^x. In biology, differentiating exponential growth and decay functions can help to analyze population growth and decay rates.