1. The problem statement, all variables and given/known data

1) If f(x)= sin^4x, then f '(pi/3)

2) Given f(x) = x/tanx, find f '(3pi/4)

3) If f(x) = sinxcosx, then f '(pi/6)

4) Differentiate: f(x) = x^2 + 2tanx

Questionthat I have answered but not sure if it's really the right answer:

5) Find the equation of the tangent line to the graph of f(x) = (x-1) / (x+1) when x = 1

2. Relevant equations

Product Rule?

- y= f x g

F'g + g'f

Chain Rule?

- y=f / g
- ((f'g) - (g'f)) / ((g^2))

Slope?

- y=mx+b

3. The attempt at a solution

1)

f(x) = sin^4x then f '(pi/3)

Derivative of sinx = cosx therefore...

I'll assume that sin^4x has the derivative of cos^4x

Now, plug in the number...

cos^4(pi/3) = 0.065

Is that right?

2)

Given f(x) = x/tanx, find f '(3pi/4)

Chain Rule:

((f'g) - (g'f)) / (g^2)

Therefore...

((1 * tanX) - (??? * X)) / ((tanx^2))

Let * be multiplication sign and the ??? to be the "I don't know".

So, I got stuck of what the derivative of "tanx". However, what I do know is that:

tanx = sinx / cosx

tanx' = cox / -sinx <-------Is that right?

If yes, then how am I suppose to make my equation by using "cox / -sinx "?

By plugging that in... I get this:

(((1 * tanX) - ((cox / -sinx) * X))) / ((tanx^2))

Then I'm really stuck on that one... I mean, if I do plug in the "pi/3" to the "x" variables then it will just be a mess. Unless that's the only way to get the answer? Or should have I used the product rule instead?

3)

If f(x) = sinxcosx, then f '(pi/6)

Product rule:

y= f x g

F'g + g'f

Therefore...

= (cosx*cosx) + (-sinx*sinx)

= (cos^2x) + (-sin^2x)

= (cos^2(pi/6)) + (-sin^2(pi/6))

= 0.633

-Let * be a multiplication sign

-Is that right?

4)

Differentiate: f(x) = x^2 + 2tanx

So, I'll just get the derivative of the equation...

2x + 2(???)

-Let ??? be "I don't know".

So, I'm stuck. I have no idea what's the derivative of tanx have. I already encountered this problem in question #3 and I assumed that it would be:

tanx = sinx / cosx

tanx' = cox / -sinx

Is that right? If yes, then I would get this equation:

2x + 2(cosx/-sinx)

Is that right? If yes, can I simplify it much more?

5)

Find the equation of the tangent line to the graph of f(x) = (x-1) / (x+1) when x = 1

y=mx+b

Chain Rule:

= 1(x+1) - 1(x-1) / (x+1)^2

= x+1 -x +1 / (x+1)^2

= 2 / (x+1)^2

= Plug in "x"

= 2 / (1+1)^2

= 2 / 4

= 1/2

slope (m) = 1/2

Now that I have the slope, I'll just get the x & y values from plugging in "1" to the equation.

y = (1-1) / (1+1)

y = 0/2

y = 0

So: x = 1 and y = 0

Then reflect on the slope equation:

y = mx+b

Plug in the numbers from what I have gotten before:

0 = 1/2(1)+b

0 - 1/2 = b

-1/2 = b

So...

y = 1/2(x) + (-1/2)

I'll multiply the whole equation by "2" to make it more neater.

2y = 2(1/2x) + 2(-1/2)

2y = x -1

Is that right?

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I know my solutions were kind of long. But I hope that you could help me. I really want to make this happen. Or at least answer the questions correctly or in a much simplified way. =)