Source: MTH 132 Calculus I, Michigan State University, Chapter 2 – Derivatives (Sections 1–3)
Difficulty: Intermediate | Prerequisites: Chapter 1 (limits, continuity, slope of secant lines)
This is the chapter where calculus properly begins. Everything in Chapter 1 about limits was building towards one idea: measuring how fast something changes at a single instant, rather than over an interval. That idea is the derivative. You will first learn to compute derivatives at specific points using the limit definition, then generalise that into a derivative function, and finally pick up the shortcut rules (power rule, product rule, quotient rule) that make differentiation fast. If you are behind, make sure you are comfortable with limit evaluation and algebraic simplification before diving in.
The derivative measures the instantaneous rate of change of a function at a point, and it equals the slope of the tangent line there. Computing derivatives from the limit definition is important conceptually but slow in practice, so a set of differentiation rules (constant, power, sum/difference, product, quotient) lets you differentiate most functions quickly.
Derivative of f at a number c
The number f'(c) = lim(h→0) [f(c+h) − f(c)] / h, provided the limit exists. An equivalent form is f'(c) = lim(x→c) [f(x) − f(c)] / (x − c).
In simple terms, the derivative at a point tells you the exact slope of the curve at that single instant, rather than an average over an interval.
Tangent line
The line through the point (c, f(c)) whose slope equals f'(c). Its equation in point-slope form is y − f(c) = f'(c)(x − c).
Think of it as the best straight-line approximation to the curve at that point.
Derivative function (f')
The function f'(x) = lim(h→0) [f(x+h) − f(x)] / h, which gives the derivative at every x where the limit exists. Also written df/dx or (d/dx)f(x) in Leibniz notation.
In simple terms, instead of computing the slope at one point at a time, f' packages all of them into a single new function.
Leibniz notation
Writing df/dx or dy/dx instead of f'(x) or y'. The derivative at x = c is written df/dx evaluated at x = c. Leibniz notation is especially useful for the chain rule because it looks like a fraction (though it is not one).
Differentiable at c
A function f is differentiable at c if f'(c) exists. It is differentiable on an open interval (a, b) if it is differentiable at every number in (a, b).
Think of it as: the curve is smooth enough at that point to have a single, well-defined tangent line.
Not differentiable (three failure cases)
A function can fail to be differentiable at a point because of a cusp or corner (left- and right-hand limits of the difference quotient disagree), a discontinuity (the function jumps or has a hole), or a vertical tangent (the slope blows up to infinity).
Differentiability implies continuity
If f is differentiable at c, then f is continuous at c. The converse is false: a function can be continuous but not differentiable (for example, f(x) = |x − 1| at x = 1).
Power rule
If f(x) = x^n for any real number n, then f'(x) = n·x^(n−1).
Think of it as: bring the exponent down as a multiplier, then reduce the exponent by one.
Constant multiple rule
(d/dx)[c·f(x)] = c·f'(x). Constants pass straight through the derivative.
Sum and difference rules
(d/dx)[f(x) ± g(x)] = f'(x) ± g'(x). You can differentiate term by term.
Product rule
(d/dx)[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x).
In simple terms, differentiate the first and multiply by the second, then add the first multiplied by the derivative of the second.
Quotient rule
(d/dx)[f(x)/g(x)] = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]^2.
Think of it as: "low d-high minus high d-low, all over the square of what's below."
The derivative of f at c is defined as f'(c) = lim(h→0) [f(c+h) − f(c)] / h.
Equivalently: f'(c) = lim(x→c) [f(x) − f(c)] / (x − c).
The expression [f(c+h) − f(c)] / h is called the difference quotient. As h shrinks to zero, the secant line becomes the tangent line.
Once you know f'(c), the tangent line at (c, f(c)) is: y − f(c) = f'(c)(x − c).
This is just point-slope form with slope = f'(c).
Write out [f(c+h) − f(c)] / h.
Expand and simplify the numerator (cancel the terms that do not involve h).
Cancel the h in the denominator.
Take the limit as h → 0.
Example: for f(x) = 2x + 5, f'(3) = lim(h→0) [(2(3+h)+5) − (2·3+5)] / h = lim(h→0) 2h/h = 2.
Example: for f(x) = √(x+1), f'(3) requires multiplying by the conjugate to clear the radical before cancelling h.
The derivative f'(c) has units of (units of f) / (units of x).
If s(t) is position in feet and t is time in seconds, then s'(t) is velocity in feet per second.
The derivative measures instantaneous rate of change, which shows up everywhere: velocity, infection rate, rate of a chemical reaction, population growth, stock price changes, acceleration.
Because the derivative is defined by a limit, it may fail to exist. At such a point the function is called not differentiable.
Example: f(x) = |x − 1| is not differentiable at x = 1 because the left-hand and right-hand difference quotients give different values (−1 and +1).
So far: f'(c) gives the derivative at one specific point.
Replace c with x and you get the derivative function: f'(x) = lim(h→0) [f(x+h) − f(x)] / h.
The derivative function gives the slope of the original curve at every x where the limit exists.
Leibniz notation: df/dx or (d/dx)f(x). At a specific point x = c, write df/dx evaluated at x = c.
The process is the same four steps as before, but you keep x as a variable throughout instead of plugging in a number.
Example: f(x) = x² + x − 8. The difference quotient simplifies to 2x + h + 1. Taking h → 0 gives f'(x) = 2x + 1.
Example: g(x) = 5/x. Combining fractions in the difference quotient and simplifying gives g'(x) = −5/x².
Case 1: Cusps and corners (jagged edges)
The derivative is a two-sided limit. If the left-hand and right-hand limits of the difference quotient disagree, the derivative does not exist. Graphically, the curve has a sharp corner.
Case 2: Discontinuity
If the function is not continuous at c, then at least one side of the difference quotient blows up or fails to produce a finite number. Key theorem: if f is differentiable at c, then f is continuous at c. The contrapositive is what matters for spotting problems: if f is not continuous at c, it cannot be differentiable there.
Case 3: Vertical tangent
The tangent line exists but is vertical, so the slope is not a finite number. Example: g(x) = x^(1/3) at x = 0.
Three guidelines for reading the derivative off a graph:
Find where the derivative is zero: these are the points where f has a horizontal tangent (peaks, troughs, flat spots).
Determine intervals where f is increasing (f' > 0) and decreasing (f' < 0).
Estimate the steepness (the magnitude of the slope) at several points within each interval to sketch the shape of f'.
If f(x) = c (a constant), then f'(x) = 0. A horizontal line has zero slope everywhere.
If f(x) = mx + b (a line), then f'(x) = m. The slope of a line is constant.
These two results follow immediately from the limit definition, but they also serve as a sanity check on any rule you use later.
Before applying the power rule, recall two rewriting conventions:
1/x^n = x^(−n)
The nth root of x = x^(1/n)
The rule itself: if f(x) = x^n for any real number n, then f'(x) = n·x^(n−1).
Examples:
p(x) = x³ → p'(x) = 3x²
f(x) = x^(0.2) → f'(x) = 0.2·x^(−0.8)
h(x) = 1/x = x^(−1) → h'(x) = −x^(−2) = −1/x²
G(x) = √x = x^(1/2) → G'(x) = (1/2)·x^(−1/2) = 1/(2√x)
For a constant c and differentiable functions f, g:
(d/dx)[c·f(x)] = c·f'(x)
(d/dx)[f(x) + g(x)] = f'(x) + g'(x)
(d/dx)[f(x) − g(x)] = f'(x) − g'(x)
These rules let you differentiate any polynomial or sum of power functions term by term.
Examples:
f(x) = −7√x → f'(x) = −7 · (1/2)x^(−1/2) = −7/(2√x)
g(x) = x² + x^(1/3) + 2x → g'(x) = 2x + (1/3)x^(−2/3) + 2
h(x) = 5/x − x^(1.7) + 4 → h'(x) = −5/x² − 1.7·x^(0.7)
(d/dx)[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x).
The derivative of a product is not the product of the derivatives. You must apply this rule whenever two non-constant expressions are multiplied together.
(d/dx)[f(x)/g(x)] = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]².
Mnemonic: "low d-high minus high d-low, over the square of what's below."
The quotient rule is needed whenever you have a ratio of two non-trivial functions. Sometimes it is easier to simplify first: for example, h(x) = (4x² − 1)/(2x + 1) factors as (2x − 1)(2x + 1)/(2x + 1) = 2x − 1, which is just a line with derivative 2.
A tangent line is horizontal where f'(x) = 0. For a polynomial like y = 2x³ + 3x² − 12x + 1, set y' = 6x² + 6x − 12 = 0, factor, and solve.
Formula | Rule |
|---|---|
f'(c) = lim(h→0) [f(c+h) − f(c)] / h | Limit definition (at a point) |
f'(x) = lim(h→0) [f(x+h) − f(x)] / h | Limit definition (function) |
y − f(c) = f'(c)(x − c) | Tangent line (point-slope form) |
f(x) = c → f'(x) = 0 | Constant rule |
f(x) = mx + b → f'(x) = m | Linear rule |
f(x) = x^n → f'(x) = n·x^(n−1) | Power rule |
(c·f)' = c·f' | Constant multiple rule |
(f ± g)' = f' ± g' | Sum / Difference rule |
(f·g)' = f'·g + f·g' | Product rule |
(f/g)' = (f'·g − f·g') / g² | Quotient rule |
Algebra reminders: 1/x^n = x^(−n), nth root of x = x^(1/n).
If s(t) gives the position of an object at time t, then s'(t) is the object's instantaneous velocity. This is the most common physical application of the derivative in this course. Beyond velocity, the derivative captures any rate of change: the rate at which a population grows, the rate at which temperature changes, or the rate at which stock prices move.
"The derivative of a product is the product of the derivatives." It is not. You must use the product rule: (fg)' = f'g + fg'.
"Differentiable and continuous mean the same thing." Differentiability is strictly stronger. Every differentiable function is continuous, but a continuous function can have corners, cusps, or vertical tangents where it is not differentiable.
"If the function is defined at a point, it must be differentiable there." Being defined (or even continuous) is not enough. Absolute-value-type kinks are the classic exam counter-example.
"The power rule only works for positive integers." It works for any real exponent: negative exponents (giving you 1/x^n), fractional exponents (roots), and even irrational exponents.
⚠️ Expect at least one problem requiring you to find the derivative from the limit definition. Do not skip this skill even though the shortcut rules are faster.
⚠️ Know how to find the equation of the tangent line at a given point: compute f'(c), then plug into point-slope form.
⚠️ Be able to identify where a function is not differentiable from a graph (corners, jumps, vertical tangents).
⚠️ Product rule and quotient rule are tested heavily. Mixing up the signs in the quotient rule (it is f'g − fg', not the other way round) is one of the most common errors.
⚠️ Rewrite roots and reciprocals as power functions before applying the power rule. Failing to do this is a frequent source of mistakes.
True or false: If f is continuous at x = 3, then f is differentiable at x = 3.
Fill in the blank: The derivative of x^7 is ___.
True or false: (d/dx)[f(x)·g(x)] = f'(x)·g'(x).
Fill in the blank: The tangent line to y = f(x) at x = c has slope ___.
True or false: A function with a vertical tangent at a point is differentiable there.
Answers: 1. False. 2. 7x^6. 3. False (product rule needed). 4. f'(c). 5. False.
Q: Use the limit definition to find f'(x) for f(x) = 3x² − 5x + 2.
A: f'(x) = lim(h→0) [3(x+h)² − 5(x+h) + 2 − (3x² − 5x + 2)] / h = lim(h→0) [6xh + 3h² − 5h] / h = lim(h→0) (6x + 3h − 5) = 6x − 5.
Q: Find the equation of the tangent line to f(x) = x³ at x = 2.
A: f(2) = 8, f'(x) = 3x², so f'(2) = 12. Tangent line: y − 8 = 12(x − 2), or y = 12x − 16.
Q: Differentiate g(x) = (x² + 1)(3x − 4).
A: Product rule: g'(x) = 2x(3x − 4) + (x² + 1)(3) = 6x² − 8x + 3x² + 3 = 9x² − 8x + 3.
Q: Differentiate h(x) = (5x)/(x² + 1).
A: Quotient rule: h'(x) = [5(x² + 1) − 5x(2x)] / (x² + 1)² = (5x² + 5 − 10x²) / (x² + 1)² = (5 − 5x²) / (x² + 1)².
Q: At which points does f(x) = |x − 2| fail to be differentiable?
A: At x = 2. The graph has a corner there, so the left-hand and right-hand derivatives disagree.
The derivative is the foundation for everything that follows in calculus. In the next sections of Chapter 2, you will use these differentiation rules to handle trigonometric functions, compositions via the chain rule, and curves defined implicitly. Later, in Chapter 3, the derivative drives the study of curve sketching, optimisation problems, and related rates. The limit definition also returns when you study the formal epsilon-delta framework in more advanced courses.
derivative, rate of change, instantaneous rate of change, slope of tangent line, difference quotient, limit definition of derivative, f prime, dy/dx, Leibniz notation, differentiable, not differentiable, cusp, corner, vertical tangent, continuity implies differentiability, power rule, constant rule, sum rule, difference rule, constant multiple rule, product rule, quotient rule, tangent line equation, point-slope form, MTH 132, Calculus I, MSU, Chapter 2