Difficulty: Intermediate | Prerequisites: Basic derivative rules (Ch. 2.1–2.3 notes), unit circle, trig identities.
This set covers Stewart sections 2.4 through 2.6. The chain rule is arguably the single most important differentiation technique: it tells you how to differentiate a composition of functions, which is nearly every function you will meet in practice. Combined with trigonometric derivatives and implicit differentiation, these tools let you handle curves defined by equations (not just explicit y = f(x) formulas) and model periodic or oscillatory phenomena.
The six trig derivatives follow from the limit lim(x->0) sin(x)/x = 1. The chain rule, d/dx[f(g(x))] = f'(g(x)) * g'(x), handles compositions ("functions inside functions"). Implicit differentiation applies the chain rule to equations not solved for y, letting you find dy/dx for curves like circles and ellipses.
Chain rule
If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). In words: differentiate the outside function, evaluated at the inside, and multiply by the derivative of the inside. Think of it as peeling layers: outer derivative times inner derivative.
Inside function / outside function
In a composition f(g(x)), g(x) is the inside function and f is the outside function. Identifying which is which is the first step in applying the chain rule.
Implicit differentiation
A technique for finding dy/dx when y is defined implicitly by an equation like x^2 + y^2 = 25, rather than explicitly as y = f(x). You differentiate both sides with respect to x, treat y as a function of x (applying the chain rule to y-terms), and solve for dy/dx.
Trigonometric derivatives
The derivatives of the six trig functions. The two fundamental ones are (sin x)' = cos x and (cos x)' = -sin x. The others follow via quotient rule.
Key trig limit
lim(x->0) sin(x)/x = 1. This is proved using the Squeeze Theorem and is the foundation for all trig derivative formulas. Only valid for x in radians.
The key limit: lim(x->0) sin(x)/x = 1 (proved via Squeeze Theorem from the unit circle). This only works in radians
From that limit: (sin x)' = cos x. Proof sketch: expand sin(x+h) using the angle addition formula, factor, and apply the key trig limits
(cos x)' = -sin x. Similar proof, or use cos x = sin(pi/2 - x) with the chain rule
The remaining four follow from the quotient rule:
(tan x)' = sec^2(x)
(cot x)' = -csc^2(x)
(sec x)' = sec(x) tan(x)
(csc x)' = -csc(x) cot(x)
Pattern to remember: the "co-" functions (cos, cot, csc) all get a negative sign in their derivatives
If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x)
In Leibniz notation: if y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx)
Procedure: identify the outside function and the inside function, differentiate the outside (leaving the inside unchanged), then multiply by the derivative of the inside
Example: d/dx[sin(x^2)] = cos(x^2) * 2x. Outside = sin, inside = x^2
Example: d/dx[(3x + 1)^5] = 5(3x + 1)^4 * 3 = 15(3x + 1)^4
Example: d/dx[sqrt(1 + x^2)] = (1/2)(1 + x^2)^(-1/2) * 2x = x / sqrt(1 + x^2)
Nested chains: for three layers like sin(cos(x^2)), apply the chain rule twice. Differentiate the outermost, then the next layer, then the innermost
The chain rule combined with the product rule and quotient rule handles virtually any explicit function
When a curve is defined by an equation like x^2 + y^2 = 25 rather than y = f(x), you cannot isolate y easily (or at all). Implicit differentiation finds dy/dx directly
Method:
Differentiate both sides of the equation with respect to x
Every time you differentiate a term containing y, apply the chain rule: d/dx[f(y)] = f'(y) * dy/dx
Solve the resulting equation for dy/dx
Example: x^2 + y^2 = 25. Differentiating: 2x + 2y(dy/dx) = 0, so dy/dx = -x/y
Example: find the tangent line to x^2 + y^2 = 25 at (3, 4). dy/dx = -3/4, so y = 4 - (3/4)(x - 3) = -(3/4)x + 25/4
The second derivative d^2y/dx^2 can be found by differentiating dy/dx again implicitly, substituting back the expression for dy/dx
Trig derivatives (radians only):
(sin x)' = cos x
(cos x)' = -sin x
(tan x)' = sec^2 x
(cot x)' = -csc^2 x
(sec x)' = sec x tan x
(csc x)' = -csc x cot x
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Implicit differentiation: differentiate both sides with respect to x, apply chain rule to y-terms, solve for dy/dx.
Trigonometric derivatives model anything periodic: sound waves, alternating current, seasonal temperature variation, circular motion. The chain rule is essential whenever one quantity depends on another that itself depends on a third, which is the norm in engineering and physics (e.g. how does pressure change with altitude, when altitude changes with time?). Implicit differentiation is used in economics (indifference curves), physics (constraint equations), and anywhere a relationship between variables is more natural as an equation than as an explicit formula.
Students often forget the negative sign in (cos x)' = -sin x. A mnemonic: the "co-" functions always pick up a minus.
Students often forget the inner derivative in the chain rule. d/dx[sin(3x)] is 3cos(3x), not cos(3x). The factor of 3 from differentiating the inside is essential.
Students sometimes treat dy/dx as if y is a constant during implicit differentiation. Every y-term needs the chain rule, producing a dy/dx factor.
Students confuse the trig derivative formulas when angles are in degrees. All the formulas above assume radians. If x is in degrees, extra conversion factors appear.
⚠️ Chain rule problems are everywhere on Calc I exams, often combined with trig derivatives and the product/quotient rules.
⚠️ Implicit differentiation problems typically ask you to find dy/dx and then the tangent line at a given point on the curve. Do not forget to plug in both x and y coordinates.
⚠️ The limit lim(x->0) sin(x)/x = 1 may be tested directly and is the foundation for deriving (sin x)' = cos x.
⚠️ Know how to handle nested chain rules: e.g. d/dx[tan(sin(x^2))].
What is (sin x)'? (cos x.)
True or false: (cos x)' = sin x. (False. It is -sin x.)
d/dx[(2x+1)^4] = ? (4(2x+1)^3 * 2 = 8(2x+1)^3.)
For x^2 + y^2 = 9, what is dy/dx? (-x/y.)
True or false: d/dx[sin(x^3)] = cos(x^3). (False. It is 3x^2 cos(x^3); you need the inner derivative.)
Q: Differentiate f(x) = sin(x) * cos(x).
A: By the product rule: f'(x) = cos(x) * cos(x) + sin(x) * (-sin(x)) = cos^2(x) - sin^2(x) = cos(2x).
Q: Differentiate g(x) = tan(3x^2 + 1).
A: Chain rule: g'(x) = sec^2(3x^2 + 1) * 6x.
Q: Find dy/dx for the curve x^3 + y^3 = 6xy.
A: Differentiating: 3x^2 + 3y^2(dy/dx) = 6y + 6x(dy/dx). Rearranging: (3y^2 - 6x)(dy/dx) = 6y - 3x^2, so dy/dx = (6y - 3x^2) / (3y^2 - 6x) = (2y - x^2) / (y^2 - 2x).
Q: Find the slope of the tangent to x^2 + y^2 = 25 at the point (3, -4).
A: dy/dx = -x/y = -3/(-4) = 3/4.
The chain rule is the backbone of related rates (Ch. 2.8): when two quantities change with time, the chain rule links their rates. It also reappears in reverse as u-substitution for integrals (Ch. 4.5). Implicit differentiation is used again in related rates and when studying curves defined parametrically (Calculus II).
trig derivatives, chain rule, implicit differentiation, sin derivative, cos derivative, tan derivative, sec derivative, composition of functions, inside function, outside function, dy/dx, implicit, unit circle, sin(x)/x limit, radians, related rates, u-substitution, MATH 132, Stewart Chapter 2, Calculus I