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Fird cbe derivative al ech o cie Icllcwing f1ucciome si4} = 33)86}Bs} = (3...

Question

Fird cbe derivative al ech o cie Icllcwing f1ucciome si4} = 33)86}Bs} = (3

Fird cbe derivative al ech o cie Icllcwing f1ucciome si4} = 3 3)86} Bs} = (3



Answers

Calculate the derivative.
$$
f(x)=3 e^{x}-x^{3}
$$

To take the derivative of our current function, we are going to use the product rule, which is first times the derivative of the second, plus second times the derivative of the first. First term is the four Y. Second term is the quantity. So that's first times the derivative of the second, which is negative two Y plus the second times the derivative of the first four to the Y. Is really just an aide to the you who's derivative is eight to the U. D'you, which is one times the natural log of the base. If I want, I can take the common for Y out four to the Y out front, leaving me with negative two Y plus two minus Y squared times natural log for.

You're taking the derivative of a quantity. We could use the quotient rule, but it's probably easier to simplify it. First, there's a Y. In the denominator, we can divide each quantity by. Why? So it would be y squared minus six, Y plus seven, and the derivative of that quantity Would be to Y -6.

Okay I'll task is to take the derivative dysfunction right here. We're gonna have a bunch of chain rules going on here because we've got this three minus two either the negative X. Inside of a cubing. And then we've even got negative X. Inside of an exponential right there. So a couple of rounds of chain rule. Let's start with the outside one fast. Okay, derivative was something cube. We know we bring that three down in front. We leave what inside the same. We dropped that power down to a two and then we chain on. We multiply by the derivative of the insight. So the derivative of the green here. Okay the director of the Green the three will go away. We'll have negative too times either the negative X. And then we're changing onto that. The derivative of the Brown. Okay the rest of the Brown who negative X. Will just be negative one. So we'll multiply that on as well. Okay well clean all this up this negative one. Times negative two will give us positive, too. That can then multiply the three in front so we'll get six either the negative X times this Prince see that's being squared and that is our derivative here.

We're going to take the derivative of this function to do so, we will use the product rule. So we'll go the first term, which is the quantity T squared plus three times the derivative of the second. Should I. E. To the T plus the second term, E. To the T times the derivative of the first, which is to T. I'll simplify this. I'll distribute the E. To the T. It's times and then reorganize it and end up with T squared. Eat of T Plus two T. E. to the T Plus three E. To the.


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