How do I find the derivative of an arctan function?

Question

The equation given is arctan (u/1+6u)

so I used the derivative of arctan (u'/1+u^2) and I plugged my formula into it and got ((1+6u)-6u/(1+6u)6^2)/1+(u/1+6u)^2 for the derivative. I tried to simplify but my answer was wrong. I'm not sure how to resolve the composed function using arctan.

Answer 1

Hi, Phil!

 

The derivative of arctan functions is a formula that should show up in the table of derivatives on the inside cover of your textbook.  You can also Google it, but for your reference, the formula is:

 

d/dx (arctan x) = 1 /(1 + x²)

 

So, in your question, x = u /(1+6u).  x² = u² / (1+6u)².

 

Using the formula, d/dx [arctan(u / 1+6u)]

=        1               

    1 +   u²      

          (1+6u)²

 

Multiply everything by (1+6u)² to get rid of the complex fraction.  This gives you:

=     (1+6u)²       

   (1+6u)² + u²

 

Simplify:

=  (1+6u)²               

   1+12u+36u² + u²

 

=    (1+6u)²   

   37u²+12u+1

 

I hope this helps!