Question

consider the following function.
f(x) = sqrt(9-x^2)
Find the derivative from the left at x = 3.
Find the derivative from the right at x = 3.

Answer 1

find the derivaite, use chain rule, and then plug in 3 for every x

Answer 2

can you show me how?

Answer 3

Let \(u=sqrt(v)\) and \(v=9-x²\). What does the chain rule tells you?

Answer 4

tell*

Answer 5

\(\frac{d}{dx}f(g(x))=f'(g(x))*g'(x))\)

Answer 6

Make it simpler yourself by writing \[(9-x ^{2})^{\frac{ 1 }{ 2 }}\]

Now you could use that derivation rule \[\frac{ d }{ dx }x ^{n}= nx ^{n-1}\]

after that you still have to multiply it with derivation of \[\frac{ d }{ dx}(9-x ^{2})\]

Answer 7

would it be 9-2x?

Answer 8

\[\frac{ d }{ dx }(9-x ^{2})\] =\[\frac{ d }{dx }9-\frac{ d }{dx }x ^{2}\] Now what is a derivative of a constant number like 9 here?

Answer 9

0

Answer 10

that means \[\frac{ d }{ dx }(9-x ^{2}) = -2x\]

Answer 11

so then how do i find the derivative from the left and right at x=3?

Answer 12

\[\lim_{x\rightarrow 3^{-}} (9-x^2)^{1/2}\]\[\lim_{x\rightarrow 3^+} (9-x^2)^{1/2}\]

Answer 13

do i just plug in 3 for both equation?

Answer 14

@ChillOut can you help me