Source: Warmup 2 – Derivative Questions | Calculus for Business, FIU
Difficulty: Introductory Prerequisites: Basic algebra, understanding of functions and function notation, familiarity with exponents (including negative and fractional exponents).
Derivatives are the central tool of differential calculus. They measure how a function's output changes as its input changes, which in a business context means things like marginal cost, marginal revenue, and rate of growth. This material covers the foundational rules you need before tackling product, quotient, and chain rules later in the course. If you can handle exponent algebra comfortably, you are ready for this.
Three rules let you differentiate most polynomial-style functions: the constant rule (derivative of a constant is 0), the power rule (bring the exponent down and reduce it by one), and the sum/constant-multiple rule (differentiate term by term, pulling constants out front). Master the algebra of rewriting roots and fractions as power expressions and these rules cover everything in this warmup.
Derivative
The rate at which a function's output changes with respect to its input. Formally, the limit of the difference quotient as the interval approaches zero.
In simple terms, it tells you how steep the curve is at any given point.
Constant function
A function whose output never changes regardless of input, e.g. f(x) = 5.
Think of it as a flat horizontal line on a graph: no slope, so the derivative is always 0.
Power rule
The rule stating that for f(x) = x^n, the derivative is f'(x) = n x^(n-1). You bring the exponent out front as a coefficient and subtract one from it.
In simple terms, "drop the power down, knock one off."
Sum rule
The derivative of a sum of functions equals the sum of their individual derivatives: d/dx [f(x) + g(x)] = f'(x) + g'(x).
Think of it as: you can differentiate each piece separately and add the results.
Constant multiple rule
A constant factor passes straight through differentiation: d/dx [k f(x)] = k f'(x).
In simple terms, constants just come along for the ride.
Derivative notation
Two standard notations for the first derivative of y = f(x): Lagrange notation f'(x) and Leibniz notation dy/dx. Both mean exactly the same thing.
Think of f'(x) as "f-prime of x" and dy/dx as "the rate of change of y with respect to x."
If f(x) = c (any constant), then f'(x) = 0.
A constant has no rate of change. The graph is a flat line, slope = 0 everywhere.
This applies to any standalone number in a function: when you differentiate term by term, constant terms vanish.
If f'(x) and g'(x) both exist and k is a constant, then:
d/dx [k f(x) + g(x)] = k f'(x) + g'(x)
You differentiate each term independently. Constants attached to a term stay attached.
This extends to any number of terms: differentiate each one on its own, then combine.
For f(x) = x^n, where n is any real number:
f'(x) = n x^(n-1)
This is the workhorse rule for polynomial differentiation.
It works for negative exponents (x^(-2)), fractional exponents (x^(1/2) = sqrt(x)), and all real exponents.
Before applying the power rule, every term must be in the form (coefficient) times x^(exponent). Common rewrites:
Roots to fractional exponents: sqrt(x) = x^(1/2), cube root of x = x^(1/3), fourth root of x^3 = x^(3/4)
Fractions with x in the denominator to negative exponents: 1/x^3 = x^(-3), 2/x^3 = 2 x^(-3)
Combinations: 12/sqrt(x) = 12 x^(-1/2)
f(x) = x^11 - 2/x^3 - 4 sqrt(x^3) - (3/7)x + 23.4
Rewrite: x^11 - 2 x^(-3) - 4 x^(3/2) - (3/7)x + 23.4
Differentiate term by term:
d/dx [x^11] = 11 x^10
d/dx [-2 x^(-3)] = -2 (-3) x^(-4) = 6 x^(-4) = 6/x^4
d/dx [-4 x^(3/2)] = -4 (3/2) x^(1/2) = -6 x^(1/2) = -6 sqrt(x)
d/dx [-(3/7)x] = -3/7
d/dx [23.4] = 0
Result: f'(x) = 11x^10 + 6/x^4 - 6 sqrt(x) - 3/7
f(x) = 12/sqrt(x) + 0.01 x^(-2) + 7x
Rewrite: 12 x^(-1/2) + (1/100) x^(-2) + 7x
Differentiate term by term:
d/dx [12 x^(-1/2)] = 12 (-1/2) x^(-3/2) = -6 x^(-3/2) = -6/(x sqrt(x))
d/dx [(1/100) x^(-2)] = (1/100)(-2) x^(-3) = -1/50 x^(-3) = -1/(50 x^3)
d/dx [7x] = 7
Result: f'(x) = -6/(x sqrt(x)) - 1/(50 x^3) + 7
Both f'(x) and dy/dx denote the first derivative of y = f(x). They are interchangeable. True or false questions on exams frequently test whether students know these are equivalent.
\frac{d}{dx}[c] = 0 \quad \text{(constant rule)}\frac{d}{dx}[k\,f(x) + g(x)] = k\,f'(x) + g'(x) \quad \text{(sum / constant multiple rule)}\frac{d}{dx}[x^n] = n\,x^{n-1} \quad \text{(power rule, } n \in \mathbb{R}\text{)}In business calculus, the derivative of a cost function gives you the marginal cost, the additional cost of producing one more unit. The derivative of a revenue function gives marginal revenue. These are the building blocks of profit optimisation, which is where this course is heading.
Physically, the derivative of a position function is velocity, and the derivative of velocity is acceleration. The same mathematical rules apply whether the context is economics, engineering, or the natural sciences.
Students often forget to rewrite roots and fractions as power expressions before differentiating. You cannot apply the power rule to sqrt(x) directly; rewrite it as x^(1/2) first.
A common error is leaving the exponent unchanged after applying the power rule. If you bring down the 3 from x^3, the new exponent must be 2, not 3.
Students sometimes think the derivative of a constant times x is zero. The derivative of 5x is 5, not 0. Only a standalone constant (no x attached) differentiates to zero.
When dealing with negative exponents, students frequently get the sign wrong. For x^(-3), the derivative is -3 x^(-4), not 3 x^(-4). The negative exponent produces an additional negative sign when brought down.
⚠️ Expect at least one question requiring you to rewrite a root or reciprocal as a power of x before differentiating.
⚠️ True/false questions on derivative notation (f'(x) vs dy/dx) appear frequently on exams and are free marks if you know they are equivalent.
⚠️ The constant rule is often tested indirectly: a function like f(x) = 3x^2 + 7 has a derivative of 6x, not 6x + 7. The "+7" must vanish.
⚠️ Show your rewrite step. Partial credit on exams typically requires evidence that you converted roots and fractions to exponent form before applying the power rule.
True or false: The derivative of 42 is 42. (False, it is 0.)
True or false: f'(x) and dy/dx are two different notations for the same thing. (True.)
Fill in the blank: The derivative of x^7 is ___ x^___. (7 x^6.)
True or false: The derivative of 5 x^3 is 5 x^2. (False, it is 15 x^2.)
Fill in the blank: Before differentiating sqrt(x), rewrite it as x^___. (x^(1/2).)
Q: What is the derivative of f(x) = x^4 - 3x^2 + 9?
A: f'(x) = 4x^3 - 6x. The constant 9 drops out.
Q: Differentiate f(x) = 6/x^2.
A: Rewrite as 6 x^(-2). Then f'(x) = 6(-2) x^(-3) = -12 x^(-3) = -12/x^3.
Q: Find the derivative of f(x) = 5 sqrt(x) + 2x.
A: Rewrite sqrt(x) as x^(1/2). Then f'(x) = 5(1/2) x^(-1/2) + 2 = 5/(2 sqrt(x)) + 2.
Q: True or false: dy/dx and f'(x) can both represent the derivative of y = f(x).
A: True. They are two standard notations for the same quantity.
Q: Differentiate f(x) = x^11 - 2/x^3 - 4 sqrt(x^3) - (3/7)x + 23.4.
A: Rewrite as x^11 - 2 x^(-3) - 4 x^(3/2) - (3/7)x + 23.4. Differentiate term by term: f'(x) = 11x^10 + 6/x^4 - 6 sqrt(x) - 3/7.
These basic differentiation rules are the foundation for every derivative technique that follows in the course. The product rule, quotient rule, and chain rule all build directly on the power rule and sum rule covered here.
In business applications, once you can differentiate polynomial and rational functions, you can find marginal cost, marginal revenue, and profit-maximising output levels, which are core topics in business calculus exams.
The concept of rewriting expressions before differentiating (roots as fractional exponents, reciprocals as negative exponents) will reappear when you study logarithmic and exponential differentiation later in the course.
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