Difficulty: Intermediate | Prerequisites: Limits and continuity study notes, algebra review (rewriting expressions in ax^n form)
The derivative is the central concept of calculus. It measures how fast a function is changing at any given point, which is exactly what businesses need when they want to know how profit, cost, or sales respond to small changes in production or spending. This section introduces the derivative first through its formal limit definition, then through the power rule shortcut that you will use for the rest of the course. Everything here depends on the limit skills from the previous sections.
The derivative f'(x) is the instantaneous rate of change of f at x, defined as the limit of a difference quotient. For most functions on this exam, you find derivatives using the power rule (bring down the exponent, subtract one). The derivative also gives the slope of the tangent line at any point, which lets you write tangent line equations and find where the tangent is horizontal.
Derivative
The instantaneous rate of change of a function at a point. Written f'(x) or dy/dx. Think of it as: "How fast is the output changing right now, per tiny change in input?"
Difference quotient
The expression [f(x+h) − f(x)] / h. It measures the average rate of change over an interval of width h. The derivative is what this becomes as h → 0.
Limit definition of the derivative
f'(x) = lim (h→0) [f(x+h) − f(x)] / h. This is the formal definition you must use when the exam says "use the definition." In simple terms, you are shrinking the interval to zero and seeing what the average rate of change settles on.
Secant line
A line through two points on a graph. Its slope is the average rate of change between those points: [f(b) − f(a)] / (b − a).
Tangent line
A line that touches the graph at exactly one point and has slope equal to the derivative there. The equation is y − f(a) = f'(a)(x − a), or equivalently y = f'(a)(x − a) + f(a).
Instantaneous rate of change
Another name for the derivative at a specific point. In business contexts, this is the marginal cost, marginal profit, or marginal revenue.
Power rule
If f(x) = x^n, then f'(x) = n·x^(n−1). This works for any real exponent, including negative and fractional powers. Think of it as: "bring down the exponent, subtract one from it."
Horizontal tangent line
A tangent line with slope zero, meaning f'(x) = 0 at that point. Finding where the tangent is horizontal means solving f'(x) = 0. These points are candidates for maxima, minima, or inflection points.
Before differentiating, you must rewrite expressions so every term looks like ax^n. This is pure algebra, not calculus, but getting it wrong ruins the derivative.
3/x² = 3x⁻²
3x²√x = 3x² · x^(1/2) = 3x^(5/2)
−5 / ⁵√(x⁷) = −5x^(−7/5)
√x / x² = x^(1/2) / x² = x^(−3/2)
The slope of the secant line through (a, f(a)) and (b, f(b)) is [f(b) − f(a)] / (b − a).
Example: f(x) = 3x². Secant through (2, 12) and (5, 75): slope = (75−12)/(5−2) = 63/3 = 21.
The slope through (a, f(a)) and (a+h, f(a+h)) is [f(a+h) − f(a)] / h.
Example: f(x) = 3x², a = 2. Slope = [3(2+h)² − 12] / h = [12+12h+3h² − 12] / h = 12 + 3h.
As h → 0, this secant slope becomes the tangent slope: m = 12.
The exam will ask you to "use the definition" for specific functions. The four steps:
Step 1: Compute f(x+h).
Step 2: Compute f(x+h) − f(x).
Step 3: Divide by h: [f(x+h) − f(x)] / h. Simplify (cancel h).
Step 4: Take the limit as h → 0. The result is f'(x).
Worked examples:
f(x) = 5. S1: 5. S2: 0. S3: 0. S4: f'(x) = 0. (Constants have derivative zero.)
f(x) = 4 − x. S1: 4−x−h. S2: −h. S3: −1. S4: f'(x) = −1.
f(x) = 3x² + 1. S1: 3(x+h)²+1 = 3x²+6xh+3h²+1. S2: 6xh+3h². S3: 6x+3h. S4: f'(x) = 6x.
f(x) = 2/x. S1: 2/(x+h). S2: 2/(x+h) − 2/x = −2h/[x(x+h)]. S3: −2/[x(x+h)]. S4: f'(x) = −2/x².
Once you have proved derivatives using the definition, the power rule is the shortcut for the rest of the course.
If y = x^n, then y' = nx^(n−1).
Constant multiple rule: d/dx [cf(x)] = c · f'(x).
Sum/difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
Derivative of a constant: d/dx [c] = 0.
Worked examples:
y = x⁷. y' = 7x⁶.
y = 10. y' = 0.
y = 1/x⁴ = x⁻⁴. y' = −4x⁻⁵ = −4/x⁵.
f(x) = (3x²)/5 − 7/(2x³) + 10. Rewrite: (3/5)x² − (7/2)x⁻³ + 10. f'(x) = (6x)/5 + 21/(2x⁴).
f(x) = ³√(x⁴) − 1/x² = x^(4/3) − x⁻². f'(x) = (4/3)x^(1/3) + 2x⁻³ = (4/3)³√x + 2/x³.
h(x) = (2x−3)(5−x). FOIL first: −2x²+13x−15. h'(x) = −4x+13 = 13−4x.
f(x) = (3x²−5)/x. Divide each term by x: 3x − 5x⁻¹. f'(x) = 3 + 5/x².
To find the equation of the tangent line at x = a:
Find f(a) (the y-coordinate of the point).
Find f'(a) (the slope of the tangent).
Use point-slope form: y = f'(a)(x − a) + f(a).
Examples:
f(x) = x² − 5 at x = −1. f(−1) = 1−5 = −4. f'(x) = 2x, so f'(−1) = −2. Tangent: y = −2(x+1) − 4 = −2x − 6.
f(x) = x⁴ at x = −2. Given f'(−2) = −32 and f(−2) = 16. Tangent: y = −32(x+2) + 16 = −32x − 48.
A horizontal tangent means the slope is zero: set f'(x) = 0 and solve for x.
f(x) = x³ − 3x² + 10. f'(x) = 3x² − 6x = 3x(x−2) = 0. Horizontal tangent at x = 0 and x = 2.
f(x) = 36/x + 4x. f'(x) = −36/x² + 4 = 0. Solve: 36/x² = 4, x² = 9, x = ±3. Horizontal tangent at x = −3 and x = 3.
Average rate of change = [f(b) − f(a)] / (b − a). In business: the average change in profit, tax, or sales over an interval.
Instantaneous rate of change = f'(a). In business: the marginal profit, marginal cost, or marginal revenue at a specific production level.
Examples:
Profit P(x) = 45x − 0.025x² − 5000. At 800 units: P(800) = $15,000. P'(x) = 45 − 0.05x, so P'(800) = $5. Interpretation: at 800 car seats, profit is $15,000 and is increasing at $5 per additional seat.
Property tax T(x) = 30x² + 50x + 400 (x years after 2006). T'(x) = 60x + 50. In 2009 (x = 3): T'(3) = $230/year. From 2006 to 2010: T(4) − T(0) = (480+200+400) − 400 = $680.
Sales S(t) = 0.03t³ + 0.5t² + 2t + 3. S'(t) = 0.09t² + t + 2. At t = 5: S(5) = $29.25 million, S'(5) = $9.25 million/month.
Limit definition of the derivative:
f'(x) = lim (h→0) [f(x+h) − f(x)] / h
Power rule:
d/dx [x^n] = n · x^(n−1), for any real n
Constant multiple rule:
d/dx [c · f(x)] = c · f'(x)
Sum/difference rule:
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
Derivative of a constant:
d/dx [c] = 0
Tangent line equation at x = a:
y = f'(a)(x − a) + f(a)
Average rate of change from a to b:
[f(b) − f(a)] / (b − a)
Horizontal tangent condition:
Set f'(x) = 0 and solve for x.
Students often forget to rewrite expressions in ax^n form before differentiating. If you try to differentiate 2/√x directly without rewriting it as 2x^(−1/2), you will almost certainly get the wrong answer.
Students often confuse the algebra step with the derivative step. On this exam, you must clearly separate them. Write f(x) = [rewritten form], then on a new line write f'(x) = [derivative]. Using f(x) when you mean f'(x) loses marks.
Students sometimes apply the power rule to products like (2x−3)(5−x) without expanding first. At this stage of the course, you do not have the product rule. You must FOIL first, then differentiate.
Students often forget that the derivative of a constant is zero, not one. d/dx [10] = 0, always.
In business application problems, students confuse average rate of change with instantaneous rate of change. Average = [f(b)−f(a)]/(b−a) over an interval. Instantaneous = f'(a) at a single point.
⚠️ "Use the definition of the derivative" questions require the full four-step limit process. You cannot use the power rule shortcut when the question specifies the definition.
⚠️ You must clearly separate the algebra (rewriting in ax^n form) from the differentiation. Use f(x) for the rewritten function and f'(x) for the derivative. Marks depend on this.
⚠️ Tangent line equations must be in slope-intercept form (y = mx + b) unless told otherwise.
⚠️ Horizontal tangent problems are a standard question: differentiate, set equal to zero, solve.
⚠️ Business word problems will ask for both the function value (e.g. profit at 800 units) and the derivative value (e.g. rate of change at 800 units), plus a written interpretation of each.
True or False: The derivative of a constant function is always 0. True.
Fill in the blank: To differentiate y = 1/x⁴, first rewrite it as y = ______. x⁻⁴.
True or False: If f'(a) = 0, the tangent line at x = a is vertical. False. A slope of zero means the tangent line is horizontal.
Fill in the blank: The average rate of change uses two points. The instantaneous rate of change uses the ______ at a single point. Derivative.
True or False: To differentiate (2x−3)(5−x), you can apply the power rule directly to each factor. False. You must expand (FOIL) first, then apply the power rule to each term.
Q: Use the definition of the derivative to find f'(x) for f(x) = 3x² + 1.
A: S1: f(x+h) = 3(x+h)²+1 = 3x²+6xh+3h²+1. S2: f(x+h)−f(x) = 6xh+3h². S3: [6xh+3h²]/h = 6x+3h. S4: lim (h→0) (6x+3h) = 6x. So f'(x) = 6x.
Q: Differentiate g(x) = 2/√x − x³/9 − 7x⁴ − 3.
A: Rewrite: g(x) = 2x^(−1/2) − x³/9 − 7x⁴ − 3. g'(x) = −1 · x^(−3/2) − x²/3 − 28x³ = −1/√(x³) − x²/3 − 28x³.
Q: Find the equation of the tangent line to f(x) = x² − 5 at x = −1.
A: f(−1) = 1−5 = −4. f'(x) = 2x, so f'(−1) = −2. Tangent: y = −2(x−(−1)) + (−4) = −2x−2−4 = −2x−6.
Q: Find the x-values where f(x) = x³ − 3x² + 10 has a horizontal tangent.
A: f'(x) = 3x² − 6x = 3x(x−2) = 0. x = 0 and x = 2.
Q: P(x) = 45x − 0.025x² − 5000 gives profit in dollars for x car seats. Find the profit and the instantaneous rate of change at x = 800, and interpret.
A: P(800) = 45(800) − 0.025(640000) − 5000 = 36000 − 16000 − 5000 = $15,000. P'(x) = 45 − 0.05x, so P'(800) = 45 − 40 = $5. At 800 car seats, the profit is $15,000 and is increasing at a rate of $5 per additional seat.
The derivative builds directly on limits: every derivative is a limit of a difference quotient. Continuity is a prerequisite for differentiability (a function must be continuous at a point to have a derivative there, though the reverse is not always true). Later in the course, derivatives will be used for optimisation (finding maximum profit, minimum cost), curve sketching, and more complex business models involving the product rule and chain rule.
Derivative, differentiation, power rule, limit definition of derivative, difference quotient, secant line, tangent line, slope of tangent, instantaneous rate of change, marginal cost, marginal profit, marginal revenue, average rate of change, horizontal tangent, f'(x), dy/dx, algebra review, ax^n form, FOIL before differentiating, MAC2233, calculus for business, FIU, exam 1 review, section 2.5