Difficulty: Intermediate | Prerequisites: Limits and Continuity (Ch. 1 notes)
This set covers Stewart sections 2.1 through 2.3: the formal definition of the derivative, the derivative as a function, and the core differentiation formulas (power, product, quotient rules). Derivatives are the first major payoff of limits. Where limits ask "what value is the function approaching?," derivatives ask "how fast is it changing?" If you are comfortable evaluating limits by factoring and cancelling 0/0, you are ready for this material.
The derivative f'(a) measures the instantaneous rate of change of f(x) at x = a, defined as the limit of the difference quotient. Once you know the power rule, sum rule, product rule, and quotient rule, you can differentiate any combination of polynomial, radical, and rational functions without returning to the limit definition each time.
Derivative
The instantaneous rate of change of f(x) at x = a, defined as f'(a) = lim(h->0) [f(a+h) - f(a)] / h. Think of it as the slope of the curve at a single point.
Differentiable
A function is differentiable at x = a when the derivative limit f'(a) exists. If the limit does not exist, the function is non-differentiable (singular) at that point. Corners, cusps, and vertical tangents are typical non-differentiable points.
Difference quotient
The expression [f(a+h) - f(a)] / h, giving the average rate of change over the interval from a to a+h. In simple terms, this is the slope of the secant line that the derivative refines into the tangent slope.
Newton notation
Writing the derivative as f'(x), f''(x) for the second derivative, and so on. Named for Isaac Newton.
Leibniz notation
Writing the derivative as df/dx, meant to evoke the ratio of infinitesimal changes. d^2f/dx^2 is the second derivative. The notation is not literally a fraction, but it behaves like one in the chain rule.
Power rule
For f(x) = x^p with any real number p, the derivative is (x^p)' = p * x^(p-1). In simple terms, bring the exponent down as a coefficient and reduce the power by one.
Product rule
(f * g)' = f' * g + f * g'. The derivative of a product is not the product of the derivatives. Think of it as: differentiate the first and leave the second alone, then leave the first alone and differentiate the second, then add.
Quotient rule
(f/g)' = [f' * g - f * g'] / g^2. Often memorised as "low d-high minus high d-low, all over low squared."
Higher derivatives
The second derivative f''(x) is the derivative of f'(x), measuring how the rate of change itself is changing. For position, the first derivative is velocity; the second is acceleration; the third is jerk.
f'(a) = lim(h->0) [f(a+h) - f(a)] / h, equivalently lim(x->a) [f(x) - f(a)] / (x - a)
The derivative gives the slope of the tangent line at (a, f(a)). Tangent line equation: y = f(a) + f'(a)(x - a)
Computing derivatives from the definition always involves a 0/0 limit. The standard approach: expand the numerator, cancel h, then evaluate
Example: f(x) = 1/(x+1), find f'(2). Combining fractions gives [1/(h+3) - 1/3] / h = -1/[3(h+3)], so f'(2) = -1/9
Example: f(x) = sqrt(x), find f'(2). Multiply by conjugate radical: [sqrt(2+h) - sqrt(2)] / h * [sqrt(2+h) + sqrt(2)] / [sqrt(2+h) + sqrt(2)] = 1/(2*sqrt(2))
Left and right derivatives: when f is defined differently on each side of a point, compute one-sided limits separately. If they agree, f'(a) exists. If they disagree, there is a corner and f is non-differentiable
Key theorem: differentiability implies continuity. If f'(a) exists, then f is continuous at a. The converse is false: f(x) = |x| is continuous at 0 but not differentiable there (corner)
The derivative function f'(x) = lim(h->0) [f(x+h) - f(x)] / h gives the slope at every point where the limit exists
Sketching f'(x) from the graph of f(x): where f has a hill or valley (horizontal tangent), f' = 0. Where f is rising, f' > 0. Where f is falling, f' < 0. Steeper slope means larger |f'|
Example: f(x) = x(x-2) = x^2 - 2x. The minimum at x = 1 gives f'(1) = 0. Algebraically, f'(x) = 2x - 2, a line through (1, 0)
Example: f(x) = x^3 - x. Hill near x = -0.6, valley near x = 0.6. f'(x) = 3x^2 - 1
Numerical derivatives: when no algebra trick works, approximate f'(a) by computing the difference quotient for small h values (e.g. h = 0.1, 0.01, 0.001) and observe the trend
Basic derivatives:
(c)' = 0 for any constant
(x)' = 1
(x^p)' = p * x^(p-1) for any real p (the power rule)
Derivative rules (assuming f and g are differentiable):
Sum: (f + g)' = f' + g'
Difference: (f - g)' = f' - g'
Constant multiple: (cf)' = c * f'
Product: (fg)' = f'g + fg'
Quotient: (f/g)' = (f'g - fg') / g^2
Product rule proof sketch: model f(x)g(x) as the area of a rectangle with sides f and g. A small change in x changes both sides, producing three strips of new area: (Delta f)g + f(Delta g) + (Delta f)*(Delta g). The third term vanishes in the limit
Strategy: for a complicated expression, decide whether to apply the product/quotient rule directly, or simplify first (multiply out, rewrite radicals as fractional powers). Both approaches should give the same answer
Example: f(x) = (5x^2 + 1)(sqrt(x) - 3). Either apply the product rule, or expand to 5x^(5/2) - 15x^2 + x^(1/2) - 3 and differentiate term by term
f''(x) = (f'(x))' = d^2f/dx^2
Physical meaning for position f(t): f'(t) = velocity, f''(t) = acceleration, f'''(t) = jerk
Example: f(t) = 16t^2 (free fall). f'(t) = 32t ft/s (velocity). f''(t) = 32 ft/s^2 (constant gravitational acceleration). f'''(t) = 0 (no jerk: gravity pulls smoothly)
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}Tangent line: y = f(a) + f'(a)(x - a)
Basic derivatives:
(c)' = 0
(x^p)' = p * x^(p-1) for any real p
Specific cases: (x^2)' = 2x, (x^3)' = 3x^2, (sqrt(x))' = 1/(2*sqrt(x)), (1/x)' = -1/x^2
Rules:
(f + g)' = f' + g'
(cf)' = c * f'
(fg)' = f'g + fg'
(f/g)' = (f'g - fg') / g^2
Higher: f''(x) = (f'(x))', position -> velocity -> acceleration -> jerk
Any quantity that changes over time has a derivative. In physics, velocity and acceleration are first and second derivatives of position. In economics, marginal cost is the derivative of total cost with respect to quantity produced. In biology, population growth rate is the derivative of population size with respect to time.
Students often think (fg)' = f' * g'. This is wrong. The product rule gives f'g + fg'. The "just multiply the derivatives" mistake is one of the most common errors on exams.
Students often confuse "not differentiable" with "not continuous." A corner point like |x| at x = 0 is continuous but not differentiable. However, a discontinuous point is always non-differentiable.
Students forget that (x^p)' = p * x^(p-1) works for negative and fractional p, not just positive integers. So (1/x)' = (x^(-1))' = -x^(-2) = -1/x^2, and (sqrt(x))' = (x^(1/2))' = (1/2)x^(-1/2).
Students sometimes apply the quotient rule when it is simpler to rewrite the function. For example, (x^5 + 1) / (x * sqrt(x)) can be rewritten as x^(7/2) + x^(-3/2) and differentiated term by term.
⚠️ Computing a derivative from the limit definition is a standard exam problem. Practise the algebra of expanding (x+h)^n and cancelling h.
⚠️ Product and quotient rule applications are tested constantly. Know both rules cold, and check your sign in the quotient rule (f'g MINUS fg').
⚠️ Sketching the derivative graph from the original graph (and vice versa) is a common visual question. Horizontal tangent on f means zero on f', and so on.
⚠️ "Show that f is differentiable at x = a" for a piecewise function requires computing both one-sided derivative limits and showing they are equal.
True or false: the derivative of a constant is always 0. (True.)
Fill in the blank: (x^7)' = ____. (7x^6.)
True or false: (fg)' = f'g'. (False. The correct rule is f'g + fg'.)
If f(t) = 16t^2, what is f''(t)? (32.)
True or false: if f has a corner at x = a, then f'(a) does not exist, but f is still continuous at a. (True.)
Q: Use the limit definition to find f'(x) for f(x) = x^2 - 3x.
A: [f(x+h) - f(x)] / h = [(x+h)^2 - 3(x+h) - x^2 + 3x] / h = [2xh + h^2 - 3h] / h = 2x + h - 3. As h -> 0, f'(x) = 2x - 3.
Q: Differentiate g(x) = (3x^2 + 1)(x^3 - 2x) using the product rule.
A: g'(x) = (6x)(x^3 - 2x) + (3x^2 + 1)(3x^2 - 2) = 6x^4 - 12x^2 + 9x^4 - 6x^2 + 3x^2 - 2 = 15x^4 - 15x^2 - 2.
Q: Differentiate h(x) = (x^2 + 1) / (x - 3) using the quotient rule.
A: h'(x) = [(2x)(x - 3) - (x^2 + 1)(1)] / (x - 3)^2 = [2x^2 - 6x - x^2 - 1] / (x - 3)^2 = (x^2 - 6x - 1) / (x - 3)^2.
Q: Find the equation of the tangent line to y = x^3 at the point (2, 8).
A: f'(x) = 3x^2, so f'(2) = 12. Tangent line: y = 8 + 12(x - 2) = 12x - 16.
Q: For f(t) = 5t^3 - 2t + 1, find the velocity and acceleration at t = 1.
A: Velocity: f'(t) = 15t^2 - 2, so f'(1) = 13. Acceleration: f''(t) = 30t, so f''(1) = 30.
The differentiation rules in this section are the building blocks for everything that follows. Trigonometric derivatives (Ch. 2.4) and the chain rule (Ch. 2.5) extend the toolkit so you can differentiate compositions and trig functions. Applications of derivatives, including optimisation, curve sketching, and related rates (Chapters 2.6 through 3), all depend on being able to differentiate quickly and accurately.
The product rule reappears in integration by parts (Calculus II), and the quotient rule is the basis for the derivative of tan(x), sec(x), and other trig ratios.
derivative definition, limit definition of derivative, difference quotient, power rule, product rule, quotient rule, differentiation, differentiable, non-differentiable, corner, cusp, tangent line, slope, Newton notation, Leibniz notation, higher derivatives, second derivative, acceleration, velocity, jerk, MATH 132, Stewart Chapter 2, Calculus I