Chapter -- Derivatives 导数

Posted 2020-08-08

1. 导数与变化率

通常，我们称曲线上某一个点切线的斜率为曲线在该点上的斜率。如果我们对着该点无限将其放大，曲线在有限的视野范围内就会变成了直线。

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 is that if we zoom in far enough toward the point, the curve looks almost like a straight line.

放大的程度越高，曲线和该点的切线就越相似。

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

 

 

2. 函数的导数

A function ƒ is differentiable at a if ƒ‘(a) exists, it is differentiable on an open interval (a,b) if it is differentiable at every number in the interval.

If ƒ is differentiable at a, then ƒ is continuous at a.

可导必然连续，但是连续未必可导。

用刚才无限区域方法的思路来看，如果ƒ在a点不可导的时候，无论怎么放大这个点，都不能看出这是一条直线。

 

 

3. 导数公式

千呼万唤始出来：

ƒ(x)=C; ƒ‘(x)=0

ƒ(x)=xⁿ; ƒ‘(x)=nxⁿ⁻¹

ƒ(x)=x⁻ⁿ; ƒ‘(x)=−nx⁻ⁿ⁻¹

ƒ(x)=a^x; ƒ‘(x)=a^xlna

ƒ(x)=e^x; ƒ‘(x)=e^x

ƒ(x)=log_ax; 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

 

导数的四则运算：

[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. 链式法则

The chain rule(链式法则):

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

F‘(x)=ƒ‘(g(x))*g‘(x)

 

The power rule combined with the chain rule

If n is any real number and μ=g(x) is differentiable, then:

 

 

5. 隐函数的导数

The functions that we have met so far can be 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, are defined implicitly by a relation between x and y such as: x²+y²=6xy.

It is not easy to solve this equation for y explicitly as a function of x by hand. But we don‘t need to solve an 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‘.

当我们在遇到上述这样的函数的时候，往往很难求解。但是，既然我们的目的是求导数，那么我们根本没有必要解出这样一个复杂的方程来。这时可以用隐函数的方法绕开解方程这样一个步骤，直接对=两边的式子求导即可。当遇到复杂的求导问题时，记得要用导数的运算法则将它们转换成简单的问题。