I keep forgetting how to do it, and I've forgotten like 5 times already. Crazy how that happens.
Let's start with \(e^x\). If we want to differentiate it, it stays the same. No change. \(e^x\) derives to \(e^x\).
Next, let's look at \(2e^x\). Again, no change. \(2e^x\) differentiates to \(2e^x\).
Now, let's look at \(e^{2x}\). Now, things will change. We just multiply the derivative of the exponent with \(e\). Everything else stays the same. \(e^{2x}\) differentiates to \(2e^{2x}\).
We might also come across \(2e^{2x}\). Here, we'll just multiply the derivative of the exponent with the coefficient of \(e\). \(2e^{2x}\) will derive to \(4e^{2x}\).
What prompted me to write this was a question on parametric differentiation. We have \(x=te^{2t}\). I thought this'd derive to \(2te^{2t}\). This is not the case. Remember, we can check all of this through the product rule, and using the product rule to check this, we would get \(t \times 2e^{2t} + e^{2t} \times 1\). We end up with \(e^{2t} + 2te^{2t}\). Whaddaya know.
Key takeaway: use the product rule when you have a product of two things.
Remember, integration is just reverse differentiation. If we know how to derivate \(e\), we know how to integrate it.
Let's start with \(e^x\). Differentiated, we would get \(e^x\). Integrated, we get \(e^x\). Why does this happen? The derivative of the exponent here is 1. Multiplying and/or divide by this value yeilds no change in our results.
Next, \(2e^x\). No change here, as was the case before.
Now, \(e^{2x}\). If this derived to \(2e^{2x}\) through the multiplication of the exponent's derivative, this will integrate to \(\frac{1}{2}e^{2x}\) through the division of the exponent's derivative.
\(2e^{2x}\). Dividing by the derivative of the exponent, we'll end up with \(e^{2x}\).
What about the example above? How would we go about integrating \(te^{2t}\)? We'll use integration by parts to figure this out, and our result will look familiar.
\(t \times 2e^{2t} - \int (1 \times 2e^{2t})\)
\(2te^{2t}-\int 2e^{2t}\)
\(2te^{2t} - e^{2t}\)
If you're ever doubtful of the credibilty of your integration, check by deriving that integral. You should end up with what you started with.
This article was written on 28/12/2023. If you have any thoughts, feel free to send me an email with them. Have a nice day!