Alevel數(shù)學(xué)干貨，微積分之鏈?zhǔn)椒▌t

Alevel數(shù)學(xué)干貨，微積分之鏈?zhǔn)椒▌t

1.The China rule (鏈?zhǔn)椒▌t)

If y=f(u) is a differentiable function of u and u=g(x) is a differentiable function of x, then y=f(g(x)) is a differentiable function of x and

or, equivalently,

如果復(fù)合函數(shù)處處可導(dǎo)，可使用鏈?zhǔn)椒▌t來(lái)進(jìn)行求導(dǎo)，外部函數(shù)的導(dǎo)數(shù)乘以內(nèi)部函數(shù)的導(dǎo)數(shù)。

常考題型解析：

Example 1

SOLUTION

As you saw earlier, you can break down this expression as follows.

Differentiation these gives

and

By the chain rule

2. Related-Rate Problems

相關(guān)變化率問(wèn)題

Differentiation with respect to different variables

對(duì)于不同變量的微分

The chain rule makes it possible to differentiate with respect to a variable which does not feature in the original expression. For example, the volume V of a sphere of radius r is given by

Differentiating this with respect to r gives the rate of change of volume with radius,

However you might be more interested in finding dv/dt, the rate of change of change of volume with time, t. To find this, you would use the chain rule:

相關(guān)變化率問(wèn)題是復(fù)合函數(shù)求導(dǎo)的應(yīng)用，例,半徑為r的球體積為

體積對(duì)半徑進(jìn)行微分，可得到體積對(duì)于半徑的變化率，

要求出體積對(duì)于時(shí)間的變化率dV/dt，可以使用鏈?zhǔn)椒▌t，通過(guò)半徑r對(duì)時(shí)間t進(jìn)行微分。

常考題型解析

Example 2