Limit theorem and derivative notes

1,Suppose that y is a function of x, say y = f(x). Derivatives tells people how sensitive the value of y is to small changes in x.We compute the derivative of f(x) by forming the difference quotient, and figure out what happens when delta x approaches 0.Suppose delta x = h, we care about what happens when h approaches 0

(f(x + h) - f(x))/h

That is the definition of the first order derivative.Generally, we use f`(x) or df/dx to denote this

f`(x) = (f(x+h) - f(x))/h where h approaches 0

2, f` or df/dx tells us how steep the graph of f(x) is

3, if we wanna figure out the value of f`(x) when h approaches 0, we have turn to the limit theorem. Common limit tricks are listed below

limit(x->k)k = k, where k is a constant

limit f(x)/g(x) = limit f(x)/limit g(x) when x approaches a given value

limit[f(x) + g(x)] = limit f(x) + limit g(x) when x approaches a given value

limit[f(x) X g(x)] = limit f(x) X limit g(x)

4,Rules for finding derivatives

A, The power rule

say f(x) = x^n. Here n is a number of any kind: integer, rational, positive, negative, even irrational, as in x^PI.

d/dx f(x) = d/dx x^n = nx^(n - 1)

B, Linearity of the derivative

d/dx cf(x) = c d/dx f(x)

d/dx [f(x) + g(x)] = d/dx f(x) + d/dx g(x)

C, The product rule

d/dx [f(x)g(x)] = f`(x)g(x) + f(x)g`(x)

D, The quotient rule

d/dx [f(x)/g(x)] = [f`(x)g(x) - f(x)g`(x)]/[g(x)]^2

E, The chain rule

d/dx [f(g(x))] = f`(g(x))g`(x), that means df/dg*dg/dx

F, Common equations

(sinx)` = cosx, (cosx)` = - sinx, (e^x) = e^x, (1/lnx)` = `/x, (a^x)` = (lna)a^x

4, Derivative & Curve Sketching

A, Maximum and minimum point

(x,f(x)) is a local maximum point if there is an interval (a,b) with a < x < b and f(x) >= f(z) for every z in (a,b); (x,f(x)) is a local minimum point if there is an interval (a,b) with a < x < b and f(x)<= f(z) for every z in (a,b). A local extremum is either a local minimum or a local maximum.

B, Fermat's Theorem: if f(x) has a local extremum at a=a and f is differentiable at a, then d/dx f(x) = 0. This implies that if a point (a, f(a)) is a maximum or minimum point, its first derivative must be 0. But if d/dx f(a) = 0, we cannot say that (a,f(a)) is a local extremum point

C, The second derivative

The first derivative test is that if the derivative changes from positive to negative at a point at which the derivative is zero, then there is a local maximum at the point, and similary for a local minimum. If d/dx f(x) changes from positive to negative, it is decreasing; this means that the derivative of f`(x), f``(x), might be negative, and if in fact f`` is negative then f` is deceasing, so there is a local maximum at the point;Similarly, if f` changes from negative to positive there is a local minimum at the point, and f` is increasing. If f`` is greater than 0 at the point, f` is definitely increasing, and so there is a local minimum.