find slope of line tangent to curve?

Equation below

Question

find slope of line tangent to curve?


Equation below

anonymous · Answer

$$\sqrt[3]{3-8x}   at (-3,3)$$

anonymous · Answer

I know to rewrite y

anonymous · Answer

put $f(x)=\sqrt[3]{3-8x}$
find $f'(x)$ and then find $f'(-3)$ which will be the slope of the tangent line

anonymous · Answer

so i guess the question is,do you know what the derivative of$f(x)=\sqrt[3]{3-8x}$ is?

anonymous · Answer

so $$f(x)=\sqrt[3]{3-8x}  = f(x)=(3-8x)^{3/2}$$

anonymous · Answer

or is it to the 2/3?

anonymous · Answer

not quite 
$$\sqrt[3]{3-8x}=(3-8x)^{\frac{1}{3}}$$

anonymous · Answer

cube root, just $\frac{1}{3}$

anonymous · Answer

$$f'(x) = \frac{ 1 }{ 3 }(3-8x)^{-2/3}-(-8)???$$   I know there's a part I'm missing

anonymous · Answer

the minus sign should be a times sign

anonymous · Answer

$$f'(x) = \frac{ 1 }{ 3 }(3-8x)^{-2/3}\times (-8)$$

anonymous · Answer

Oh so that's it - the sign is wrong? I'm not missing anything els from the derivative?

anonymous · Answer

chain rule, you have to multiply by the derivative of $3-8x$ which is $-8$
now to evaluate this, get rid of the exponential notation and write is as 
$$f'(x)=-\frac{8}{3\sqrt[3]{(3-8x)^2}}$$

anonymous · Answer

so I plug -3 into the derivative and get my answer

$$f'(x)=\frac{ 8 }{ 3\sqrt[3]{(3-8(-3))^{2}} }$$

$$f'(x)=\frac{ -8 }{ 3\sqrt[3]{729} }$$

what do I do with $$\sqrt[3]{729}$$
How would I inout that into my calculator to solve?

anonymous · Answer

i would take the cube root first, square last 
then you don't need a calculator

anonymous · Answer

$3-8\times (-3)=27$  
$\sqrt[3]{27}=3$
$3^2=9$

anonymous · Answer

which means of course that $\sqrt[3]{729}=9$ as well if you use a calculator but it is not necessary for this problem

anonymous · Answer

This is a review question and it says that the answer is -8/27

anonymous · Answer

yes, that is what we will get right?

anonymous · Answer

Yes - I forgot about the remaining 3 in the denominator

anonymous · Answer

you have 
$$\sqrt[3]{3-8(-3)}=9$$ so 
$$-\frac{8}{3\sqrt[3]{3-8(-3)}}=-\frac{8}{3\times 9}=-\frac{8}{27}$$

anonymous · Answer

Thanks -  my brain is sluggish

anonymous · Answer

lol mine too 
yw

anonymous · Answer

too little sleep and nowhere near enough caffeine

anonymous · Answer

good point
i will go get another cup of coffee now