# Math Help - Several questions I didn't get

1. ## A couple questions I didn't get

I did several pages of calculus for homework, and these are the questions I got stuck on:

1. Find the derivative of y=2/√x + x/√3 + 6(x)^1/3.

2. Find the derivative at x= -2 for y=3u²+2u and u = √(x²+5).

3. The amount of pollution in a certain lake is P(t) = (t^0.25 + 3)³, where t is years and P is parts per million. What is the rate of change after 16 years (I know I use the limit formula but the exponents and t+h are confusing me).

4. Find the x-intercept of the function f(x)= 2x^(5/3) - 5x^(2/3).

5. If f is a differentiable function, find an expression for the derivative of h(x)=2xf(x).

Thanks!

2. Originally Posted by NAPA55
1. Find the derivative of y=2/√x + x/√3 + 6(x)^1/3.
Do you know how to apply the power rule for derivatives?

$f(x)=ax^n$
$f'(x)=nax^{n-1}$

$f(x)=\frac{2}{\sqrt{x}} + \frac{x}{\sqrt{3}} + 6x^{\frac{1}{3}}$

Convert all you radicals to exponents

$f(x)=2x^{\frac{1}{2}} + \frac{x}{\sqrt{3}} + 6x^{\frac{1}{3}}$

Now, I will leave the derivation up to you!

3. 2. Find the derivative at x= -2 for y=3u²+2u and u = √(x²+5).
using the chain rule:

$\begin{gathered}
\frac{{dy}}
{{dx}} = 6u\frac{{du}}
{{dx}} + 2\frac{{du}}
{{dx}} = \left( {6\sqrt {x^2 + 5} + 2} \right)\frac{{du}}
{{dx}} = \hfill \\
\hfill \\
= \left( {6\sqrt {x^2 + 5} + 2} \right)\left[ {x\left( {x^2 + 5} \right)^{ - 0.5} } \right] \hfill \\
\end{gathered}
$

4. Originally Posted by NAPA55
3. The amount of pollution in a certain lake is P(t) = (t^0.25 + 3)³, where t is years and P is parts per million. What is the rate of change after 16 years (I know I use the limit formula but the exponents and t+h are confusing me).
You don't need the limit "formula". Just find the derivative, and solve for when time is equal to 16.

$P'(t)=3(t^{0.25}+3)^2 (0.25t^{-0.75})$

$P'(16)=3(25)+ (\frac{1}{32}) \Rightarrow \frac{75}{32}$

Check my arithmetic, but the derivative is correct.