170- Quiz 4

Quiz 4 is here. Please remember that the first midterm is this Wednesday.

Solutions follow.

Problem 1 asks for the derivative of the function 6/\root 3\of {x^5}.

To solve this, remember the following laws of exponents:

  • \root 3\of t=t^{1/3},
  • (t^a)^b=t^{ab}, and
  • 1/t^a=t^{-a}.

Using these laws we quickly obtain that 6/\root 3\of {x^5}=6/x^{5/3}=6x^{-5/3}. Now, to take the derivative of this function, we apply two of the rules covered on lecture:

  • (cf)'=cf' when c is a constant, and
  • (x^k)'=kx^{k-1} when k is a constant.

We have (6x^{-5/3})'=6\cdot(-5/3)\cdot x^{-8/3}=-10x^{-8/3}.

Problem 2 asks to use the product rule to find the derivative of (x^2+5x-3)(x^5 -6x^3+3x^2-7x+1).

The product rule says that

(fg)'=f'g+fg',

and in this case it gives us that the derivative is

(2x+5)(x^5-6x^3+3x^2-7x+1) +(x^2+5x-3)(5x^4-18x^2+6x-7).

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