Chapter (3)--derivatives derivative

1. Derivative and rate of change

In general, we say that the slope of a point tangent on a curve is the slope of the curve at that point. If we enlarge the point infinitely, the curve will become a straight line within the limited field of view.

We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. The idea was that if we have zoom in far enough toward the point, the curve looks almost like a straight line.

The higher the magnification, the more similar the curve is to the tangent of the point.

The more we have zoom in, the more the parabola looks like a line. In and words, the curve becomes almost indistinguishable from its tangent line.

2. Derivative of the function

A functionƒis differentiable at a ifƒ ' (a) exists, it's differentiable on an open interval (a, b) if it is differentiabl E at every number in the interval.

Ifƒis differentiable at a, thenƒis continuous at a.

may lead to a continuous, but continuous may not be conductive.

Using the idea of the infinite region method, if ƒ is not at point A, no matter how to enlarge the point, it can not be seen as a straight line.

3. Derivative formula

Awaited starts out:

ƒ (x) =c; ƒ ' (x) =0

ƒ (x) =xn; ƒ ' (x) =nxn−1

ƒ (x) =x−n; ƒ ' (x) =−nx−n−1

ƒ (x) =ax; ƒ ' (x) =axlna

ƒ (x) =ex; ƒ ' (x) =ex

ƒ (x) =logax; ƒ ' (x) =1/xlna

ƒ (x) =lnx; ƒ ' (x) =1/x

ƒ (x) =sinx; ƒ ' (x) =cosx

ƒ (x) =cosx; ƒ ' (x) =−sinx

ƒ (x) =tanx; ƒ ' (x) =sec2x

ƒ (x) =cotx; ƒ ' (x) =−csc2x

Arithmetic of derivative:

[ƒ (x) +g (x)] ' =ƒ ' (x) +g ' (x)

[ƒ (x) −g (x)] ' =ƒ ' (x) −g ' (x)

[ƒ (x) *g (x)] =ƒ ' (x) *g (x) +ƒ (x) *g ' (x)

[ƒ (x)/g (x)] = [ƒ ' (x) *g (x) −ƒ (x) *g ' (x)]/[G (x)]2

4. Chain rules

The chain rule (chain rule):

If g is differentiable at x Andƒis differentiable at g (x), then the composite function F=ƒ°g defined by F (x) =ƒ (g (x)) is Differentiable at x and F ' are given by the product:

F ' (x) =ƒ ' (g (x)) *g ' (x)

The power rule combined with the chain rule

If N is a real number andμ=g (x) is differentiable and then:

5. Derivative of the implicit function

The functions that we had met so far can is described by expressing one variable explicitly in terms of another variable- -for example, y=x3+1, or in general, y=ƒ (x). Some functions, however, is defined implicitly by a relation between x and y such as:x2+y2=6xy.

It is the not-easy-to-solve this equation for y explicitly as a function of the x by hand. But we don ' t need to solve a equation for y in terms of x in order to find the derivative of Y. Instead We can use the method of implicit differentiation. This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for Y '.

When we encounter such a function, it is often difficult to solve. However, since our goal is to take the derivative number, then we simply do not need to solve such a complex equation. At this point, we can use the method of implicit function around the Kaixie equation to take a direct derivative of both sides of the formula. When encountering complex derivation problems, remember to use derivative algorithms to convert them into simple questions.

Chapter (3)--derivatives derivative