Differentiate problem

[hank]

I need to differentiate:

\(\displaystyle \frac{{4x}}{{\sqrt {4x^2 - 1} }}\)

Here's what I get:

\(\displaystyle \frac{{\left( {4x^2 - 1} \right)^{\frac{1}{2}} 4 - 4x\frac{1}{2}\left( {4x^2 - 1} \right)^{\frac{{ - 1}}{2}} }}{{4x^2 - 1}}\)

Multiply that by this to get rid of fractions:
\(\displaystyle \frac{{4x^2 - 1}}{{4x^2 - 1}}\)

Gets me to here:

\(\displaystyle \frac{{4\left( {4x^2 - 1} \right)^{\frac{3}{2}} - 2x}}{{\left( {4x^2 - 1} \right)^2 }}\)

I'm not sure where to go from this point.

Am I ok up to here, and if so, where should I go from here?

 

[galactus]

Product rule:

\(\displaystyle \L\\4x(4x^{2}-1)^{\frac{-1}{2}}\)

\(\displaystyle \L\\4x(\frac{-1}{2})(4x^{2}-1)^{\frac{-3}{2}}(8x)+(4x^{2}-1)^{\frac{-1}{2}}(4)\)

\(\displaystyle \L\\\frac{-16x^{2}}{(4x^{2}-1)^{\frac{3}{2}}}+\frac{4}{(4x^{2}-1)^{\frac{1}{2}}}\)

Multiply right side by \(\displaystyle (4x^{2}-1)\)

\(\displaystyle \L\\\frac{-16x^{2}}{(4x^{2}-1)^{\frac{3}{2}}}+\frac{4}{(4x^{2}-1)^{\frac{1}{2}}}\cdot\frac{(4x^{2}-1)}{(4x^{2}-1)}\)

\(\displaystyle \H\\\fbox{\frac{-4}{(4x^{2}-1)^{\frac{3}{2}}}}\)

 

[hank]

Thanks tons, I see where I made my mistake!

 

[soroban]

Hello, Hank!

Galactus is absolutely correct . . .

Differentiate: \(\displaystyle f(x) \:=\:\L\frac{4x}{\sqrt{4x^2\,-\,1}}\)

Here's what I get:

\(\displaystyle \L f'(x) \;=\;\frac{{\left( {4x^2 - 1} \right)^{\frac{1}{2}}\cdot4\, - \,4x\cdot\frac{1}{2}\cdot\left(4x^2 - 1 \right)^{-\frac{1}{2}}}}{4x^2 - 1}
\\) \(\displaystyle \nwarrow\) times the derivative of \(\displaystyle 4x^2-1\)

If there is a constant coefficient, I prefer to "leave it out front".

We have: \(\displaystyle \L\:f(x)\;=\;4\,\cdot\,\frac{x}{\left(4x^2-1\right)^{1/2}}\)

Then: \(\displaystyle \L f'(x) \;= \;4\,\left[\frac{(4x^2-1)^{\frac{1}{2}}\cdot1 \,- \,x\cdot\frac{1}{2}\cdot\left(4x^2-1\right)^{-\frac{1}{2}}\cdot8x}{4x^2-1} \right]\)

. . . . . . . . . . .\(\displaystyle \L=\;4\,\left[\frac{(4x^2-1)^{\frac{1}{2}} \,-\,4x^2(4x^2-1)^{-\frac{1}{2}}}{4x^2-1}\right]\)


Multiply top and bottom by \(\displaystyle (4x^2 - 1)^{\frac{1}{2}}\)

\(\displaystyle \L f'(x) \;= \;\frac{(4x^2-1)^{\frac{1}{2}}}{(4x^2-1)^{\frac{1}{2}}}\,\cdot\,4\left[\frac{(4x^2-1)^{\frac{1}{2}} \,- \,4x^2(4x^2-1)^{-\frac{1}{2}}}{4x^2-1}\right]\)


. . . . . . \(\displaystyle \L=\;4\left[\frac{4x^2\,-\,1\,-\,4x^2}{(4x^2-1)^{\frac{3}{2}}}\right] \;= \;\fbox{\frac{-4}{\left(4x^2\,-\,1\right)^{\frac{3}{2}}}}\)


[The "box" is for you, Cody!]