Calculus Volume 1 · free preview
The Chain Rule
The chain rule differentiates compositions by multiplying the derivatives of the layers, the indispensable rule behind implicit differentiation, related rates, and every composite model.
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.
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