Calculus 1000-Derivative of Trig Functions (Sec 007, October 19, 2026)

Overview

Asghar Ghorbanpour reviews core differentiation rules, then derives the sine derivative from the limit definition using the identities for sin(x + h) and the special limit sin(h)/h = 1. He builds the trigonometric derivative table, applies it with product and quotient rules, finds a tangent line for e^x sec x at (0,1), and introduces limits that can be evaluated by rewriting them to match sin(u)/u.

Key takeaways

The fundamental trig derivatives are sin'(x) = cos x and cos'(x) = -sin x; the sine result follows from the addition identity and lim(h→0) sin(h)/h = 1.
Expressing tan x as sin x/cos x and applying the quotient rule yields tan'(x) = sec²x through the identity sin²x + cos²x = 1.
The complete derivative table used here includes cot'(x) = -csc²x, sec'(x) = sec x tan x, and csc'(x) = -csc x cot x.
For y = e^x sec x, the product rule gives slope 1 at x = 0; since the graph passes through (0,1), its tangent line is y = x + 1.
To evaluate lim(x→0) sin(3x)/x, create the standard ratio sin(3x)/(3x) and factor out 3, giving a limit of 3.
Writing out each product- or quotient-rule step makes errors easier to locate, especially when differentiating nested sums of products.

Chapters

0:00 Review of Sum, Product, Quotient, and Constant-Multiple Rules

The sum and difference rules differentiate each function separately, then add or subtract the results.
The constant-multiple rule keeps the constant factor outside the derivative.
For a product f(x)g(x), use f'(x)g(x) + f(x)g'(x), not the product of the derivatives.
For a quotient f(x)/g(x), use [f'(x)g(x) - f(x)g'(x)]/g(x)^2, provided the denominator is nonzero.

4:00 Power Rule, Basic Derivatives, and the Exponential Function

The power rule is d(x^n)/dx = n x^(n-1), with real exponents where the expression is defined.
The derivative of a constant is 0; the examples 1/x and √x come from exponents -1 and 1/2.
The derivative of e^x is e^x, a rule Ghorbanpour uses in later product-rule examples.
Ghorbanpour's near-term goal is to extend the differentiation table so students can quickly differentiate the functions covered in class.

8:00 Setting Up the Derivative of sin x from the Limit Definition

Ghorbanpour introduces sin x and cos x as the two foundational trigonometric derivatives, since other trig functions can be expressed using them.
For sin x, the derivative definition gives lim(h→0) [sin(x+h) - sin x]/h.
The sine addition identity, sin(x+h) = sin x cos h + cos x sin h, separates the expression into familiar limits.
During this limit, x is fixed while h approaches 0, so sin x and cos x act as constants.

11:00 Using sin(h)/h to Prove the Derivative of Sine

Splitting the difference quotient produces sin x·lim(h→0)[cos h - 1]/h + cos x·lim(h→0) sin h/h.
The special limit lim(h→0) sin h/h = 1 supplies the second term.
To evaluate [cos h - 1]/h, multiply by the conjugate expression and use sin²h + cos²h = 1 to reduce it to a limit involving sin h/h.
The remaining limit is 0, giving d(sin x)/dx = cos x.

23:00 Cosine, Tangent, and the Six Trigonometric Derivative Rules

The derivative of cos x is -sin x; Ghorbanpour attributes the negative sign to the cosine angle-addition identity.
Writing tan x as sin x/cos x and applying the quotient rule gives tan'(x) = (cos²x + sin²x)/cos²x = sec²x.
The class table includes cot'(x) = -csc²x, sec'(x) = sec x tan x, and csc'(x) = -csc x cot x.
Ghorbanpour recommends deriving the quotient-based formulas once using the identities, then memorizing them for efficient differentiation.

29:00 Practice Problems with Secant, Tangent, Sine, and Exponential Terms

The practice set includes h(x) = sec x/(1 + tan x) and g(x) = tan x/(x² + 2x).
A third product-and-sum example uses t sin t together with an e^t cos t term, requiring the product rule twice.
Students have two minutes to work through the derivatives, using the rule table and checking their steps.
Ghorbanpour says simplification is not required for this exercise, emphasizing correct rule application.

33:00 Applying Product and Quotient Rules to the Trig Practice Set

For the function with t sin t and e^t cos t, differentiate each product separately, then add the results.
The product involving t sin t gives sin t + t cos t; differentiating e^t cos t gives e^t cos t - e^t sin t.
For h(x) = sec x/(1 + tan x), the quotient-rule numerator is sec x tan x(1 + tan x) - sec x sec²x.
For g(x) = tan x/(x² + 2x), the numerator is sec²x(x² + 2x) - tan x(2x + 2), over (x² + 2x)²; Ghorbanpour encourages writing steps clearly to catch errors.

38:00 Tangent Line to e^x sec x at (0,1)

A tangent line needs a point on the graph and a slope; for y = f(x), the slope at x₀ is f'(x₀).
At x = 0, e^0 sec 0 = 1, so the point on y = e^x sec x is (0,1).
The product rule gives f'(x) = e^x sec x + e^x sec x tan x, and evaluating at 0 gives slope 1.
Using point-slope form, the tangent line is y - 1 = 1(x - 0), or y = x + 1.

44:00 Rewriting Trigonometric Limits to Match sin(u)/u

The limit sin(u)/u approaches 1 as u approaches 0, which also works when u is a more complicated expression.
For lim(x→0) sin(3x)/x, rewrite it as 3·sin(3x)/(3x) to match the standard limit and obtain 3.
Ghorbanpour assigns a further limit involving sin(x - 2)/(x² + x - 6) and asks students to transform it into the same standard form.
Factoring x² + x - 6 as (x - 2)(x + 3) reveals how the denominator's matching factor can be used with the sine limit.

Transcript excerpt

0:01 Is that recording? Okay, sure. >> [clears throat] >> So today we are continuing on differentiation. So you you have gone through the the definition of the differentiation. Probably we will need to do some some more things, but I want to push the sections to the end of where we are supposed to be and then hopefully we will have one two sessions before the midterm so that we can go through some more examples. But then you learned about the the rules differentiation rules, the basic ones. That's what we need at this point. Now we are we are adding more to this toolbox. So what I'm assuming you know so far are

0:51 differentiation rules. Which is okay, the sum and the difference rule which is what? Which is saying that you have two functions both of them differen- differentiable. So I'm not writing the details. I'm just writing the rules here. If you want to find the derivative of the sum or difference, you do differentiate each one and then add them or subtract them. And then there is the rules. There is constant multiple rule which says if you have a constant

1:41 mul- constant multiplied to a function the the the this whole new function is differentiable and the derivative is you just differentiate the function and multiply that constant. There is product rule. This is important and this is basically maybe the first in the first first time you look at it, it may seem like, "Okay, something is wrong with this rule." So far we were doing the limit laws, everything, and product of limits of product was product of limits. But when it comes to differentiation,

2:31 it is not just the derivative. It is the quotient of the differences. So that means that you're not going to have the

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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.