Differentiate
f (x)= [(6x + x^5)^-1 + x]^2
f (x)= cos[(3x) / (x+1)]
f (x)= sqrt(x) + 1/4sin (2x)^2
f (x)= sin(cos x)

Question

Differentiate 
f (x)= [(6x + x^5)^-1 + x]^2
f (x)= cos[(3x) / (x+1)]
f (x)= sqrt(x) + 1/4sin (2x)^2
f (x)= sin(cos x)

anonymous · Answer

No one wants to come here to explicitly do your homework. Use the quotient rule, chain rule, and product rule to solve all of these.
One example, with f(x) = sin(cosx)
The chain rule states that you take the derivative of the whole function multiplied by the derivative of the inside., so
$$f(x) = sin(cosx)$$
$$f'(x) = cos(cosx)*-sinx$$
$$f'(x) = -sinxcos(cosx)$$

anonymous · Answer

No one has to come here to do my homework. I simply want to know how to go about doing the questions. This was my last resort, because I am desperate. Thanks for your time, negativity is not needed.

anonymous · Answer

Sorry if i came off negative. I'll try to help you learn, just letting you know no one really wants to just put out answers for you. Did the explanation I gave make sense? Do you still need help with the other?

anonymous · Answer

I understand what you mean, it wouldn't be fair for someone to just give me the answers. I actually want to know how to do the work. The explanation you gave did make sense, looking at the questions I didn't realize that I needed to use the product rule and quotient rule. However the first one do I have to use the chain rule twice? If you know what I mean.

anonymous · Answer

Yup, you'll have to do the derivative of the whole thing, then the inside, and when doing the inside do the chain rule again, you're right. 

The second one, also, you'll do the chain rule, but when multiplying by the derivative of the inside, you'll need to do the quotient rule. If you wanna check your answers here feel free to and i'll check them out.

anonymous · Answer

I would really appreciate that, I did the second one and I got (-sin(3x)/(x+1))((x+1)(3)-(3))

anonymous · Answer

Almost, the first part is right, but remember the quotient rule is $$\frac{ (Lo*dHi)-(Hi*dLo) }{ (Lo)^{2} }$$

So, it'll be

$$\frac{ ((x+1)(3))-((3x)(1)) }{ (x+1)^{2} } = \frac{ 3 }{ (x+1)^{2} }$$

anonymous · Answer

Ohhhh, righttt. Thank you so much.

anonymous · Answer

No problem