Steps for Solving Differential Quotient Equations in Calculus

[Ryuk1990]

I don't take a Calculus class(I'm learning on my own), but I'm just curious as to what are the steps to solving the following equations.

Homework Statement

These are the following problems that I'm having trouble solving.

y(x) = cos(x)

y(x) = [tex]\sqrt{x}[/tex]

y(x) = sin(x)

y(x) = x[tex]^{n}[/tex]

Homework Equations

For all of these, the limit of [tex]\Delta[/tex]X approaches 0.

The Attempt at a Solution

The problem is, I can't attempt it because it's different from other ones. I can do simpler things like y(x) = 9x^2 no problem. These are different though.

 

[n!kofeyn]

y(x) = cos(x)

I'm guessing that you are needing to find y'(x) in each case using the definition of the derivative, which is the limit of the difference quotient.

I'll help you with the first one, and maybe that will get you started. You need to know this special limit to do the first one:
[tex]\lim_{x\to0} \frac{1 - cos x}{x} = 0[/tex]
You also need the trig identity:
[tex]\cos(u+v) = \cos u\cos v - \sin u\sin v[/itex]

Then the derivative is:
[tex] y'(x) = \lim_{h\to 0} \frac{\cos(x+h) - \cos x}{x} [/itex]
Now apply the above information to compute y'(x).