How do I differentiate this?

Question

steve816 · Answer

$$f(t)=e^{t+2}$$

zepdrix · Answer

You start with your rule for differentiating e^x,
and follow it up with chain rule.

zepdrix · Answer

$$\large\rm \frac{d}{dx}e^{x}=e^{x}$$Therefore,$$\large\rm \frac{d}{dt}e^{t+2}=e^{t+2}\frac{d}{dt}(t+2)$$Chain rule, ya?

zepdrix · Answer

Here is another option!!
If you apply an exponent rule,$$\large\rm e^{t+2}=e^t\cdot e^2$$But.. e^2 is just some number. It's not changing, it's just constant.
You can think of it like this,$$\large\rm C e^t$$
just some constant in front of the e^t.$$\large\rm \left(Ce^t\right)'=C\left(e^t\right)'=Ce^t$$And from there, remember that constant coefficients ignore the differentiation process.
They just come along for the ride.$$\large\rm \left(e^2\cdot e^t\right)'=e^2\left(e^t\right)'=e^2\cdot e^t=e^{t+2}$$

zepdrix · Answer

That second approach might be a little more confusing though ^
What do you think Stevey?

steve816 · Answer

Wow, very clearly explained, thank you! I don't think my teacher went over the chain rule yet, so I didn't know how to do these, but thanks!

zepdrix · Answer

Chain rule is the most complicated of your simple differentiation rules.
Once you do learn it, practice practice practice!

It's the one rule that can cause you a lot of trouble if you're not super strong with it.

steve816 · Answer

Alright, thanks for the advice :)