Calculus 1000- Chain Rulel I (Sec 007, October 20, 2026)

Overview

Asghar Ghorbanpour reviews limits built from the foundational limit sin(t)/t as t approaches 0, then introduces the chain rule for differentiating composite functions. Examples with f(x)=1/x, a quadratic g, and function-value tables show how to identify inner and outer functions, while the final section extends familiar power, reciprocal, and square-root rules to functions of u.

Key takeaways

To evaluate limits involving sin(h(x))/h(x) as h(x) approaches 0, rewrite the expression to expose sin(t)/t and track the remaining factors.
The chain rule is (f∘g)'(x)=g'(x)f'(g(x)); the outer derivative must be evaluated at the inner function's output.
For table-based composite derivative problems, first find the inner function's value at the requested point, then use it to look up the outer derivative.
For any differentiable u(x), the power rule becomes d(uⁿ)/dx=n uⁿ⁻¹u', the reciprocal rule becomes d(1/u)/dx=−u'/u², and the square-root rule becomes d(√u)/dx=u'/(2√u).
The table examples yield (f∘g)'(0)=−8, (f∘g)'(1)=0, and (f∘f)'(1)=6 using the supplied values of f, g, f', and g'.

Chapters

0:00 Building sin(t)/t from Limits Involving sin(5x)

The target pattern is sin(t)/t when t approaches 0, using the known limit value 1.
For a limit involving sin(5x), multiply and divide by 5x to create sin(5x)/(5x).
Since 5x approaches 0 as x approaches 0, the standard sine limit applies and the remaining factor determines the result.

7:15 Applying the Sine Limit to Shifted and Scaled Arguments

For sin(x−2)/((x−2)(x+3)) as x approaches 2, separate sin(x−2)/(x−2) from 1/(x+3).
Setting t=x−2 changes the first limit to sin(t)/t as t approaches 0; direct substitution gives the other factor as 1/5.
For a ratio involving sin(5x) and sin(7x), insert matching factors 5x and 7x to obtain the ratio 5/7.

17:13 The Chain Rule for Compositions f∘g

The chain rule applies when the inner function g is differentiable at a and the outer function f is differentiable at g(a).
For a composition, (f∘g)'(x)=g'(x)f'(g(x)): differentiate the inner function, then multiply by the outer derivative evaluated at the inner function's value.
At a specific point a, the formula becomes g'(a)f'(g(a)), with the outer derivative evaluated at g(a), not automatically at a.

21:30 Differentiating f∘g, g∘f, and g∘g

Using f(x)=1/x, Ghorbanpour finds f'(x)=−1/x² and substitutes g(x) into that derivative for f'(g(x)).
For g∘f, the inner derivative is f'(x), while the outer derivative g' must be evaluated at f(x).
For g∘g, both functions are g, but the outer derivative still takes g(x) as its input; the lecture's quadratic notation is inconsistent in places, so the substitution pattern is the reliable focus.

32:14 Using Function Tables to Evaluate Composite Derivatives

A table gives f(0)=1, f(1)=2, f(2)=0 and f'(0)=−1, f'(1)=3, f'(2)=2.
It also gives g(0)=2, g(1)=2, g(2)=1 and g'(0)=−4, g'(1)=0, g'(2)=−5.
The exercises ask for (f∘g)'(0), (f∘g)'(1), and (f∘f)'(1), requiring both function values and derivative values from the table.

36:26 Evaluating Composite Derivatives from the Tables

For (f∘g)'(0), use g'(0)=−4 and f'(g(0))=f'(2)=2, giving −8.
For (f∘g)'(1), g'(1)=0, so the composite derivative is 0 regardless of the outer derivative value.
For (f∘f)'(1), use f'(1)=3 and f'(f(1))=f'(2)=2, giving 6.

40:15 Writing the Chain Rule as a Rule for Functions of u

When u is a function of x, differentiating f(u) requires multiplying f'(u) by u'.
The power rule becomes d(uⁿ)/dx = n uⁿ⁻¹u', combining the ordinary power rule with the chain rule.
This notation extends familiar differentiation rules to expressions whose inputs are functions rather than just x.

44:30 Extending Reciprocal and Square-Root Rules to Composite Functions

For a square root of a function, the derivative is u'/(2√u), applying the square-root rule and multiplying by the inner derivative.
For 1/u, the derivative is −u'/u²; the negative sign is explicitly corrected during the example.
The closing examples combine chain rule with prior rules, including differentiating expressions involving sin(x) and constants.

46:20 Practice Plan for Combining Differentiation Rules

Ghorbanpour emphasizes practicing the chain rule together with power, reciprocal, square-root, and other previously learned rules.
The class will continue updating its differentiation table and apply the rules to more composite expressions.
The session ends at about 47:53, with further chain-rule practice planned for the next class.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Asghar Ghorbanpour.