Calculus Volume 1 · 免费预览
链式法则
链式法则通过将各层的导数相乘来对复合函数求导，它是隐函数求导、相关变化率以及一切复合模型背后不可或缺的法则。
1. Composing rates
Composing rates
2. The rule in action
The rule in action
3. Combining with the other rules
Combining with the other rules
4. Why the rule holds
Why the rule holds
5. Chain rule statement
If y = f(u) and u = g(x), which equation states the chain rule correctly?
6. Derivative of a composite power
What is the derivative of h(x) = (3x^2 + 1)^5?
7. Three-layer chain
For y = sqrt(sin(x^2)), which factorization of dy/dx is correct?
8. Product with an inner chain
What is the derivative of h(x) = x^2 sin(3x)?
9. Identifying the outermost operation
To differentiate h(x) = (x sin x)^3, which rule applies at the top level first?
10. Leibniz reading
The chain rule in Leibniz notation reads dy/dx = dy/du times du/dx. What is the correct interpretation of this notation?
11. Derivative of e to the kx
The function h(x) = e^(5x) has inner u = 5x. What is h prime of x? Answer like 5e^(5x).
12. Derivative of a shifted power
What is the derivative of h(x) = (x + 1)^4? Answer like 4(x + 1)^3.
13. Inner derivative factor
Differentiating sin(2x) by the chain rule produces cos(2x) times a constant. What constant? Answer with a number.
14. Chain through a square
What is the derivative of h(x) = (x^2)^2 evaluated and simplified? Answer like 4x^3.