Source: MATH 104 Calculus 1, Penn State
Tags: product rule, chain rule, higher derivatives, analytic function, Taylor series, composition of functions, nth derivative
Difficulty: Intermediate | Prerequisites: Basic differentiation (power rule, trig derivatives), understanding of function composition. For the analytic functions section, some familiarity with sequences and series is helpful.
Once you know how to differentiate basic functions, the next question is: how do you differentiate combinations of functions? The product rule and chain rule are the two main tools for this job. Higher derivatives extend the idea by asking what happens when you differentiate again and again. Analytic functions tie everything back to series, connecting the derivative machinery to Taylor and power series representations. These concepts appear throughout MATH 104 and carry into every applied maths and science course you will take afterward.
The product rule handles derivatives of f(x)·g(x). The chain rule handles derivatives of f(g(x)). Higher derivatives are what you get when you keep differentiating. Analytic functions are the special class of functions that equal their own Taylor series in a neighbourhood of every point in their domain.
Product rule
The differentiation rule for a product of two functions: (f·g)' = f'·g + f·g'. Think of it as: "derivative of the first times the second, plus the first times the derivative of the second."
Chain rule
The differentiation rule for a composition of functions: if F(x) = f(g(x)), then F'(x) = f'(g(x)) · g'(x). In simple terms, you differentiate the outer function (leaving the inner function alone), then multiply by the derivative of the inner function.
Higher derivatives
The result of differentiating a function more than once. The second derivative f''(x) is the derivative of f'(x), the third derivative f'''(x) is the derivative of f''(x), and so on. The nth derivative is written f⁽ⁿ⁾(x).
Analytic function
A function that can be represented by a convergent power series (Taylor series) in a neighbourhood of every point in its domain. Think of it as a function that is so well-behaved it equals its own Taylor expansion everywhere locally.
Power series
An infinite series of the form Σ aₙ(x − x₀)ⁿ. In simple terms, it is an infinite polynomial centred at the point x₀.
Taylor series
A specific power series built from the derivatives of a function at a point: Σ [f⁽ⁿ⁾(x₀)/n!] · (x − x₀)ⁿ. Think of it as the best possible polynomial approximation of a function near x₀.
Used when differentiating f(x) · g(x)
Formula: (f·g)' = f'·g + f·g'
The mnemonic many students use: "first times derivative of second, plus second times derivative of first"
Extends to three or more factors, though that is less common on exams: (f·g·h)' = f'·g·h + f·g'·h + f·g·h'
Used when differentiating a composition f(g(x))
Formula: F'(x) = f'(g(x)) · g'(x)
Outer derivative evaluated at the inner function, multiplied by the inner derivative
In Leibniz notation: if y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx)
The chain rule is the single most-used derivative technique beyond the basic rules. Nearly every applied differentiation problem involves it
The second derivative f''(x) tells you about concavity (concave up vs concave down)
The third derivative f'''(x) is sometimes called the "jerk" in physics (rate of change of acceleration)
For n ≥ 4, notation switches to f⁽ⁿ⁾(x) because primes become hard to count
The nth derivative is the derivative of the (n−1)th derivative
Not every function has derivatives of all orders; when a function does, it is called infinitely differentiable (or smooth)
A function is analytic at x₀ if it equals its Taylor series in some neighbourhood of x₀
Being infinitely differentiable is necessary but not sufficient for analyticity
There exist smooth functions that are not analytic (e.g. f(x) = e^(−1/x²) at x = 0)
Real analytic and complex analytic functions are related categories, but complex analytic functions have stronger properties
A function is analytic on its domain if and only if its Taylor series about x₀ converges to the function in some neighbourhood for every x₀ in that domain
Product Rule
(f · g)' = f' · g + f · g'
Chain Rule
(f ∘ g)'(x) = f'(g(x)) · g'(x)
Or equivalently, if y = f(u) and u = g(x):
dy/dx = (dy/du) · (du/dx)
Higher Derivatives (notation)
First: f'(x) or df/dx
Second: f''(x) or d²f/dx²
Third: f'''(x) or d³f/dx³
nth: f⁽ⁿ⁾(x) or dⁿf/dxⁿ
Taylor Series
f(x) = Σₙ₌₀^∞ [f⁽ⁿ⁾(x₀) / n!] · (x − x₀)ⁿ
The product rule comes up whenever two quantities that each vary are being multiplied, for example, revenue = price × quantity, where both price and quantity change over time. The chain rule appears whenever one measurement depends on another that itself depends on a third: a doctor tracking how blood pressure changes with dosage, which in turn changes with time. Higher derivatives matter in engineering and physics: the second derivative of position is acceleration, and the third is jerk, relevant in designing smooth elevator rides and roller coasters.
Students often write the product rule as (f·g)' = f'·g'. This is wrong. You cannot just differentiate each factor separately and multiply. Both terms (f'·g and f·g') are needed.
With the chain rule, a frequent mistake is forgetting to multiply by the inner derivative g'(x). Students differentiate the outer function but drop the second factor.
Students sometimes confuse the product rule and the chain rule. The product rule is for f(x) · g(x). The chain rule is for f(g(x)). Multiplication is not composition.
The source PDF writes the product rule formula using the ∘ (composition) symbol, i.e. (f∘g)' = f'∘g + f∘g'. This is a notational error in the source. The product rule uses ordinary multiplication (f·g), not composition. Do not let this confuse you.
⚠️ Product rule and chain rule problems appear on virtually every MATH 104 exam. You need to be able to apply both quickly and correctly.
⚠️ Combination problems are common: "differentiate x² · sin(3x)" requires the product rule on the outside and the chain rule on sin(3x). Layered application is typical.
⚠️ Higher derivatives questions may ask you to find f''(x) or f'''(x) for a given function, or to identify concavity using the second derivative.
⚠️ Analytic functions and Taylor series tend to appear later in the course. Expect conceptual questions about what it means for a function to be analytic and the convergence condition.
True or false: (f·g)' = f'·g'.
Fill in the blank: If F(x) = f(g(x)), then F'(x) = f'(___) · g'(x).
True or false: The third derivative of f is the derivative of the second derivative.
Fill in the blank: A function is analytic at x₀ if it equals its ___ series in some neighbourhood of x₀.
True or false: Every infinitely differentiable function is analytic.
Answers: 1. False. 2. g(x). 3. True. 4. Taylor. 5. False.
Q: Differentiate h(x) = x³ · cos(x) using the product rule.
A: h'(x) = 3x² · cos(x) + x³ · (−sin(x)) = 3x² cos(x) − x³ sin(x).
Q: Differentiate F(x) = sin(5x²) using the chain rule.
A: The outer function is sin(u), whose derivative is cos(u). The inner function is u = 5x², whose derivative is 10x. So F'(x) = cos(5x²) · 10x = 10x cos(5x²).
Q: Find the second derivative of f(x) = e^(2x).
A: f'(x) = 2e^(2x) (chain rule). f''(x) = 4e^(2x) (chain rule again).
Q: What does it mean for a function to be analytic at a point x₀?
A: It means the function equals its Taylor series expansion about x₀ in some neighbourhood of that point. The Taylor series must converge to the function, not just exist.
Q: Differentiate y = (3x + 1)⁴ · e^x. Which rules do you need?
A: Both the product rule and the chain rule. Product rule gives: y' = [(3x+1)⁴]' · e^x + (3x+1)⁴ · e^x. For the first term, chain rule gives [(3x+1)⁴]' = 4(3x+1)³ · 3 = 12(3x+1)³. So y' = 12(3x+1)³ · e^x + (3x+1)⁴ · e^x.
The chain rule is essential for FTC Part 1 problems where the upper limit of integration is itself a function of x (e.g. d/dx ∫₀^(x²) f(t) dt). The product rule comes back in integration by parts, which is the reverse of the product rule. Higher derivatives feed directly into Taylor series, since the Taylor coefficients are built from f⁽ⁿ⁾(x₀). If you are heading into differential equations, all of these concepts are foundational.
product rule, chain rule, higher derivatives, second derivative, third derivative, nth derivative, analytic function, Taylor series, power series, Maclaurin series, composition of functions, Leibniz notation, smooth function, infinitely differentiable, convergent series, radius of convergence, MATH 104, calculus 1, Penn State, derivative rules