# Homework Help: Use the Approximate Relationship to Prove

1. Sep 22, 2013

### MelissaJL

1. The problem statement, all variables and given/known data
Use the approximate relationship to prove:
Δf~$\frac{df}{dx}$Δx

a) $\frac{dx^{n}}{dx}$= nxn-1

b)$\frac{dcos(θ)}{dθ}$=-sin(θ)

2. Relevant equations
a)N/a?? I'm not sure if I need any other equations than the one given.

b) sin(ε)~ε and cos(ε)~1 when ε<<1.
3. The attempt at a solution
a) So I'm honestly quite lost and know that my attempt is going to be far off.
I thought maybe I could sub in the values into the given equation:

f2-f1~df/dx (x2-x1) (because they are deltas)
xn-x~$\frac{dx^{n}}{dx}$(Δn)
xn-1~$\frac{dx^{n}}{dx}$(Δn)

But now I have no clue what I'm doing again and I know what I'm doing doesn't make much sense. I honestly haven't even attempted part (b) because I don't understand what to do with part (a). Just to be clear I'm not looking for help on part (b) yet until I try to attempt the problem. I'm just looking for help with part (a) so that I can try to do part (b) afterwards. Can someone help explain to me how I am suppose to use that formula?

2. Sep 22, 2013

### fzero

For part a we have

$$\Delta f = f(x+\epsilon) - f(x) = (x+\epsilon)^n - x^n.$$

Now you want to expand $(x+\epsilon)^n$ and then take the limit that $\epsilon \rightarrow 0$. There's an identity from precalc that you can use here. For part b, you will have to expand $\cos(x+\epsilon)$ using trig identities.