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Solve the given problems by finding the appropriate derivatives.Find the slope of a line tangent to the curve of the function $y=(3 x+4)(1-4 x)$ at the point $(2,-70) .$ Do not multiply the factors together before taking the derivative. Use the derivative evaluation feature of a calculator to check your result.

Calculus 1 / AB

Chapter 23

The Derivative

Section 6

Derivatives of Products and Quotients of Functions

Derivatives

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Our goal in the problem is to evaluate dy dx at the ordered pair too negative 70. That's my little notation there and were given why equals the product three x plus four. So that quantity times 1 -4 x. So you can see that we definitely have the product rule in here. And to do the the product rule you start with the derivative of one side. I like to do the derivative of the left side which is three. That's using the power rule. You leave the right side alone and then minus now you leave the left side alone And you multiply by the derivative of the right side -4. Now you could clean this up if you wanted to but there's really no reason to. So all I need to do next is plug into and for these two exes. And that's easy for me to do because I'll leave the three alone and four times two is 81 minus eight is negative seven and then three times to six plus four is 10 times negative four. So we're looking at negative 21 plus 40 which will give me derivative leave alone. Oh, it's a plus in there. My bed. Uh huh. Of course. The I've been doing this quotient rule for a little bit and forgot what I was doing. Uh Yeah, it's addition in between the two pieces. The product rule is derivative lee loan plus. Leave alone take the derivative stereo. So this stays -40. And so we're getting more negative. We get an answer -61. So it's important that you double check your work Um and -61. Is that correct slope at that point?

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