Question

Use the limit definition of the derivative.
f(x)=1/x+3
f'(6)=?

Answer 1

\[f'(x)=\lim_{x\to 6}\frac{\frac{1}{x+3}-\frac{1}{9}}{x-6}\]

Answer 2

A derivative of a function f(x) is defined as the limit of a ratio of two expressions as x approaches a value c, in your case c=6.

Answer 3

then a bunch of algebra
\[\frac{1}{x+3}-\frac{1}{9}=\frac{9-(x+3)}{9(x+3)}=\frac{6-x}{9(x+3)}\]

Answer 4

divide by \(x-6\) to give you
\[-\frac{1}{9(x+3)}\] and finally replace \(x\) by \(6\) for your answer

Answer 5

I think a general form of the definition of a derivative should be helpful.

Answer 6

I am confused. i dont get the first line of the equation why its( 1/x+3-1/9)/x-6?

Answer 7

and the answer is -(1/81)?