# Arctangent once again

#### legacyofpiracy

##### Junior Member
Ack these arctangents keep throwing me off. I keep trying to find the derivative of these two equations and each time the computer program (online math homework) says they are wrong. If someone could please show me what I am missing it would be greatly appreciated.

Using the formula of the derivative of an arctan : (1/(1+u)^2)*(du/dx)

I tried to find the derivative of this equation:

f(x)=2arctan(8x^7)

i ended up getting this

2*(1/(1+(8x^7))^2)*(56x^6)*(7x+7)

also for the equation f(x)=tan^-1(cos(8x))

i ended up getting

(1/1+(cos(8x))^2)*(-sin(8x))+8

there really wasnt any work to show..i simply followed the formula of the arctan and took the derivative of the u. Still..any guesses where I might have gone wrong?

#### soroban

##### Elite Member
Hello, legacyofpiracy!

Using the formula of the derivative of arctan: $$\displaystyle \L \frac{1}{1 + u^2}\cdot\frac{du}{dx}$$

I tried to find the derivative of this function: .$$\displaystyle f(x)\:=\:2\arctan(8x^7)$$

I ended up getting this: .$$\displaystyle \L 2\cdot \frac{1}{1+(8x^7)^2}\cdot(56x^6)\cdot(\underbrace{7x+7})$$
. . . . . . . . . . . . . . . . . . . . . . . . . .?

also for the function: .$$\displaystyle f(x)\:=\:\tan^{-1}(\cos 8x)$$

I ended up getting: .$$\displaystyle \L\frac{1}{1+\cos^28x}\cdot(-\sin 8x) \underbrace{+ 8}$$
. . . . . . . . . . . . . . . . . . . . .?

#### Gene

##### Senior Member
You're close. Did you get the formula over the phone?
d(tan<sup>-1</sup>(u)) = 1/(1+u²)du not 1/(1+u)²

Also d(8x^7)=56x^6
Lose the (7x+7)

Try them with that.

#### legacyofpiracy

##### Junior Member
Both of you are amazing I have no idea why i wanted to keep taking the derivative of 56x^6 and cos8x.

I got the first one right, however it still keeps saying that the derivative of the function f(x)=tan^-1(cos8x) is wrong.

here is how i typed it in:

(1/(1+(cos(8x))^2))*(-sin(8x))

the trouble with these online homeworks is that you need to have every single parenthasies or else it is wrong. >.< am i just typing it in incorrectly?

-thank you both so much again

#### Gene

##### Senior Member
Close again
d(cos(8x))=-sin(8x)d(8x)=-8sin(8x)dx
Calculators are just as fussy. A necessary () omitted gives an entirely diferent problem and answer.

I'm glad Soroban and I agreed. I looked before I started typing and I was all alone. I think he cheats and types with the fingers on both hands.