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Problem 2 Easy Difficulty

Differentiate each of the following:
a. $y=e^{3 x}$
b. $s=e^{3 t-5}$
c. $y=2 e^{10 t}$
d. $y=e^{-3 x}$
e. $y=e^{5-6 x+x^{2}}$
f. $y=e^{\sqrt{x}}$

Answer

a. $3 e^{3 x}$
b. $3 e^{3 t-5}$
c. $20 e^{10 t}$
d. $-3 e^{-3 x}$
e. $(-6+2 x) e^{5-6 x+x^{2}}$
f. $\frac{1}{2 \sqrt{x}} e^{\sqrt{x}}$

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Video Transcript

{'transcript': "problem number 27 Section 45 Again, Everything in this section deals with finding derivatives of functions that involve logarithms. So here I have. Why is equal to e to the five X divided by the natural log of three x each of the five x over the natural log of three to the X. So again, guiding principle here I've got something that looks like y is equal to F over G. Why prime is quotient rule F prime G minus G prime f over cheese squared. In my particular case, eat of the five x represents F and natural law. Go three x represents a g. So to go find this derivative. So why Prime is going to be equal to F Prime G. So the derivative eat of the five X is five e to the five X so that's F primes, Then times G. That's gonna be times natural Log of three x minus G prime f So g prime f. That is gonna be what one over three x times 32 That's G prime. All of that times F, which is either the five x over G squared, which is Ellen of three x squared and it looks like I can simplify a bit by multiplying everything by five X. And so I multiply everything by five acts she meet motive, everything by X because what you're going to see here? I've got three over three I can keep from having a complex fraction if I multiply everything so by X so I'm gonna have five x e to the five x natural log of three X and then minus E to the five x over X natural log of three X and then all of that is squared and it looks like I could probably factor out and eat to the five X, um, for a further simplification. So if I were to do that, I'm gonna have e to the five X and then I'm left with five x natural log of three X minus one, all of that divided by X natural log of three x squared. So each of the five x five x of natural log of three X minus one over X natural log of three x All that quantity squared"}