Chapter 4:Derivatives

Question 1

Derivative

Answer 1

The meaning of derivative is to evaluate the rate of change in a certain amount of time(delta x) of f(x) as time is denoted as x.
Therefore dervative of f at x, denoted by f’(x), is defined as:
f’(x0)=lim (f’(x0+deltax) - f(x0))/deltax as deltax tend to 0. ( if the limit exists ofc)
We call f’(x0) as the instantaneous change of function f at x0.

Question 2

Differentiability

Answer 2

As long as there is a f’(x0) at x0 then f is differentiable at x0.It is differentiable on open interval {a,b} when f(x) is differentiable with any x belongs to {a,b}.

Question 3

One - sided derivatives

Answer 3

+Left-hand derivative:
lim (f(x0+deltax) - f(x0))/deltax as deltax tend to 0-
+Right-hand derivative:
lim (f(x0+deltax) - f(x0))/deltax as deltax tend to 0+
THEOREM:
+ f’(x0) exists as long as f’(x0-) = f’(x0+) (these two derivative has to also exists)
+ if f’(x0) exists then f is** continuous **at x0. The reverse is false.

Question 4

Higher Derivatives

Answer 4

-The nth derivative: d^nf/dx^n=(d/(dx))(d^(n-1)f/dx^n-1) or f^(n)(x)={f^(n-1)(x)}’ with n=2,3,…
Note: f^(0)(x)=f(x)/ (n) here is the number of times the function derivative
Basic Formulas
1.(e^(ax+b))(n)=(a^n)(e^(ax+b))
2.(a^x)(n)=(a^x)((lna)^n)
3.(sin(ax+b))(n)=(a^n)sin(ax+b+npi/2)
4.(cos(ax+b))(n)=(a^n)cos(ax+b+npi/2)
5.(1/(ax+b))(n)={(-1^n)xn!xa^n}/(ax+b)^(n+1)
6.{(ax+b)^k}(n)=k(k-1)……(k-n+1){(ax+b)^(k-n)}xa^n with k doesn’t belong to Z
Leibneiz Formula:
(u.v)(n)=n-singma-k=0 Ckn.u(k).v(n-k)