## 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

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