Use the Approximate Relationship to Prove

[MelissaJL]

Homework Statement

Use the approximate relationship to prove:
Δf~\frac{df}{dx}Δx

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

b)\frac{dcos(θ)}{dθ}=-sin(θ)

Homework 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.

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)
x^n-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?

 

[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.