trouble finding derivatives

I am having trouble solving these problems, and finding the derivatives. I do not need them solved, but just explain the next step (unless otherwise noted):

1) sin(2x)*cos(2x)

' = sin(2x)(-2sin2x)+cos2x(__2__cos(2x))

- where did the two come from? wouldn't it just be cos(2x)?

2) cot(x)/sin(x) = cos(x)/sin^2(x)

- shouldn't there be a (-) since in the identity, there is one?

3) 4sec^2(x)

- Alright, this one, I do not quite understand what they did. It looks like they took the derivative of the number (they get 8) times the sec, then times the derivative of sec^2.

Re: trouble finding derivatives

Quote:

Originally Posted by

**droidus** I am having trouble solving these problems, and finding the derivatives. I do not need them solved, but just explain the next step (unless otherwise noted):

1) sin(2x)*cos(2x)

' = sin(2x)(-2sin2x)+cos2x(__2__cos(2x))

- where did the two come from? wouldn't it just be cos(2x)?

You are using the **chain rule**. Differentiate $\displaystyle \sin()$ then $\displaystyle 2x.$

Re: trouble finding derivatives

Quote:

Originally Posted by

**droidus** I am having trouble solving these problems, and finding the derivatives. I do not need them solved, but just explain the next step (unless otherwise noted):

1) sin(2x)*cos(2x)

' = sin(2x)(-2sin2x)+cos2x(__2__cos(2x))

- where did the two come from? wouldn't it just be cos(2x)?

The same place the "2" in the first term came from- the "2x" in the trig function. The derivative of sin(u), with respect to u, is cos(u) and if u= 2x, du/dx= 2.

$\displaystyle \frac{d(sin(2x)}{dx}= \frac{d(sin(u))}{du}\frac{du}{dx}= cos(u)(2)= 2cos(2x)$

Quote:

2) cot(x)/sin(x) = cos(x)/sin^2(x)

- shouldn't there be a (-) since in the identity, there is one?

No, why should there be? cot(x)= cos(x)/sin(x) and that sin(x) already in the denominator gives cos(x)/sin^2(x).

(You may be thinking of the "-" in the derivative of cot(x). There is NO derivative here, just an identity. If you are asking about the derivative of cot(x)/sin(x), you haven't differentiated yet. What does the quotient rule give for the derivative of cos(x)/sin^2(x)?

Quote:

3) 4sec^2(x)

- Alright, this one, I do not quite understand what they did. It looks like they took the derivative of the number (they get 8) times the sec, then times the derivative of sec^2.

No, they did not take the derivative of any "number", they used the chain rule. Letting u= sec(x), this function is $\displaystyle 4u^2$, which has derivative, with respect to u, of 4(2u)= 8u= 8 sec(u). Now multiply that by the derivative of u with respect to x, the derivative of sec(x).

Re: trouble finding derivatives

oh, ok - see for the first one... thanks!