Question

f(x)=sqrt(3x+1), find the derivative at a

Answer 1

You can f(x) as :
\[(3x+1)^{\frac{ 1 }{ 2 }}\]Still the same meaning

Answer 2

there's a short cut but I'll show you the proper one
\[\lim_{\Delta x \rightarrow 0 } \frac{ f(x+\Delta x)-f(x) }{ \Delta x }\]

Answer 3

\[\frac{ \sqrt{3(x+\Delta x) + 1} -\sqrt{3x+1} }{ \Delta x }\]

Answer 4

f(x)=sqrt(3x+1)
f'(x)=[(1/2)/sqrt(3x+1)] 3 chain rule

Answer 5

Multiply by the conjugate

Answer 6

\[\frac{ [3(x+\Delta x) +1] - 3x+1 }{ \Delta x (\sqrt{3(x+\Delta x)+1}+\sqrt{3x+1)} }\]

Answer 7

\[ 3x + 3 \Delta x +1 -3x -1 \] on the numerator
and same thing on the denominator

Answer 8

also, my bad, it's -3x -1

Answer 9

after you cancel out everything, 3 Delta x remains,
you simplify and cancel out delta x, and you'll left out with
\[\frac{ 3 }{ \sqrt{3x+3\Delta x +1}+\sqrt{3x+1} }\]

Answer 10

and since Delta x is approaching 0, you plug in 0 and you'll get
3/2sqrt(3x+1)

Answer 11

Nice development muskie, good to know procedure but why reinvent the wheel.

Answer 12

I just went over the basic steps. and Thank you for your compliment.

Answer 13

There's a shortcut of course.