Difficulty: Beginner to Intermediate Prerequisites: Familiarity with limits, basic algebra, and trigonometric functions. If limits are shaky, review those notes first.
Derivatives sit at the heart of calculus. They give you a precise way to measure how a function changes at any given point, which is the foundation for everything that follows: optimisation, curve sketching, related rates, differential equations. If limits told you what a function approaches, derivatives tell you how fast it is getting there. You need a solid handle on limit evaluation and basic function algebra before this material will land properly.
The derivative of a function at a point is the instantaneous rate of change at that point, defined formally as a limit. A handful of rules (power, product, quotient, chain, sum/difference) let you compute derivatives mechanically without returning to the limit definition each time.
Derivative
The instantaneous rate of change of a function f(x) at a point. Formally, it is the limit of the difference quotient as the interval shrinks to zero. Think of it as the slope of the curve at a single point, not over an interval.
Difference quotient
The expression [f(x) − f(a)] / (x − a). It measures the average rate of change between two points on a curve. In simple terms, it is rise over run between two points, and the derivative is what happens when you push those two points together until they merge.
Tangent line
The straight line that just touches the graph of f(x) at a single point and has the same slope as the curve at that point. Think of it as the best straight-line approximation of the curve right at that spot.
Instantaneous rate of change
The rate at which a function's output changes at one specific input value, as opposed to an average over an interval. In simple terms, it is speed at a single instant rather than average speed over a journey.
Power Rule
The differentiation rule stating that if f(x) = xⁿ, then f′(x) = nxⁿ⁻¹. Think of it as "bring the exponent down as a multiplier, then reduce the exponent by one."
Product Rule
The rule for differentiating a product of two functions: if f(x) = u(x) · v(x), then f′(x) = u′(x) · v(x) + u(x) · v′(x). In simple terms, differentiate one factor at a time while leaving the other alone, then add the results.
Quotient Rule
The rule for differentiating a fraction of two functions: if f(x) = u(x)/v(x), then f′(x) = [u′(x) · v(x) − u(x) · v′(x)] / [v(x)]². Think of it as "low d-high minus high d-low, all over low squared."
Chain Rule
The rule for differentiating composite functions: if f(x) = g(h(x)), then f′(x) = g′(h(x)) · h′(x). In simple terms, differentiate the outer function first (leaving the inner function untouched), then multiply by the derivative of the inner function.
Sum and Difference Rule
Differentiation distributes over addition and subtraction: the derivative of a sum is the sum of the derivatives, and likewise for differences. Think of it as "you can differentiate term by term."
The derivative of f(x) at the point x = a is defined as:
f′(a) = lim [f(x) − f(a)] / (x − a), as x → a
This limit, when it exists, gives the slope of the tangent line to the graph of f at x = a.
If the limit does not exist at a particular point, the function is not differentiable there.
If f(x) = xⁿ, then f′(x) = nxⁿ⁻¹.
Works for any real exponent n, including negative and fractional powers.
Example: f(x) = x⁴ gives f′(x) = 4x³.
Constant terms (like −1) vanish because the derivative of any constant is 0.
If f(x) = u(x) · v(x), then f′(x) = u′(x) · v(x) + u(x) · v′(x).
You never just multiply the individual derivatives together. The product rule exists precisely because differentiation does not distribute over multiplication.
Example: f(x) = 3x² · sin(x)
u(x) = 3x², so u′(x) = 6x
v(x) = sin(x), so v′(x) = cos(x)
f′(x) = 6x · sin(x) + 3x² · cos(x)
If f(x) = u(x)/v(x), then f′(x) = [u′(x) · v(x) − u(x) · v′(x)] / [v(x)]².
Note the subtraction in the numerator: order matters. It is always (derivative of top × bottom) minus (top × derivative of bottom).
Example: f(x) = (x² + 1)/(x − 1)
u′(x) = 2x, v′(x) = 1
f′(x) = [(2x)(x − 1) − (x² + 1)(1)] / (x − 1)²
Simplifies to (x² − 2x − 1) / (x − 1)²
If f(x) = g(h(x)), then f′(x) = g′(h(x)) · h′(x).
Think "outside-in": differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.
Example: f(x) = sin(2x)
Outer function: sin(·), derivative is cos(·)
Inner function: 2x, derivative is 2
f′(x) = cos(2x) · 2 = 2cos(2x)
If f(x) = u(x) ± v(x), then f′(x) = u′(x) ± v′(x).
Differentiation is linear: you can handle each term independently.
Example: f(x) = x⁴ − 3x² + 2x − 1
f′(x) = 4x³ − 6x + 2
Rule | Formula |
|---|---|
Power Rule | f(x) = xⁿ → f′(x) = nxⁿ⁻¹ |
Product Rule | (uv)′ = u′v + uv′ |
Quotient Rule | (u/v)′ = (u′v − uv′) / v² |
Chain Rule | [g(h(x))]′ = g′(h(x)) · h′(x) |
Sum/Difference | (u ± v)′ = u′ ± v′ |
Constant Rule | c′ = 0 |
Derivatives measure rates of change, which makes them fundamental to physics (velocity is the derivative of position, acceleration is the derivative of velocity), economics (marginal cost is the derivative of total cost), and engineering (stress-strain analysis, control systems). Any time you want to know "how fast is this quantity changing right now," you are asking for a derivative.
Students often believe you can differentiate a product by simply multiplying the derivatives of the factors. You cannot. The product rule exists because differentiation does not distribute over multiplication.
The order of subtraction in the quotient rule matters. Swapping the two terms in the numerator gives you the wrong sign, and there is no shortcut to remembering this except practice.
When applying the chain rule, students frequently forget to multiply by the derivative of the inner function. If you differentiate sin(2x) and get cos(2x) without the factor of 2, you have dropped the chain rule step.
A common error with the power rule is forgetting that the derivative of a constant term is zero, leading students to carry constants through unchanged.
⚠️ The limit definition of the derivative appears on most early calculus exams. Know how to set up and evaluate the difference quotient from scratch.
⚠️ Product rule and quotient rule applications are standard exam fare. Be ready to apply them to combinations of polynomial and trigonometric functions.
⚠️ Chain rule questions are often embedded inside product or quotient rule problems. Nested applications (chain within product, for instance) are a favourite way to test whether students can layer the rules correctly.
⚠️ Simplification after applying the quotient rule is frequently required for full marks. Do not leave the answer in expanded form unless told otherwise.
True or False: The derivative of a constant is 1. Answer: False. The derivative of a constant is 0.
Fill in the blank: If f(x) = x⁷, then f′(x) = ________. Answer: 7x⁶
True or False: The product rule states that (uv)′ = u′ · v′. Answer: False. The product rule states (uv)′ = u′v + uv′.
Fill in the blank: The chain rule says that d/dx [g(h(x))] = g′(h(x)) · ________. Answer: h′(x)
True or False: The derivative of sin(3x) is cos(3x). Answer: False. It is 3cos(3x), because the chain rule requires multiplying by the derivative of the inner function.
Q: Using the limit definition, what does f′(a) equal?
A: f′(a) = lim [f(x) − f(a)] / (x − a) as x → a.
Q: Find the derivative of f(x) = 5x³ − 2x + 7.
A: f′(x) = 15x² − 2. The power rule gives 15x² and −2; the constant 7 drops out.
Q: Differentiate f(x) = x² · cos(x).
A: Using the product rule: f′(x) = 2x · cos(x) + x² · (−sin(x)) = 2x cos(x) − x² sin(x).
Q: Differentiate f(x) = (x³ + 2) / (x − 4).
A: Using the quotient rule: f′(x) = [3x²(x − 4) − (x³ + 2)(1)] / (x − 4)² = (2x³ − 12x² − 2) / (x − 4)².
Q: Differentiate f(x) = cos(5x²).
A: Chain rule with outer function cos(·) and inner function 5x²: f′(x) = −sin(5x²) · 10x = −10x sin(5x²).
This material connects directly to integration (the reverse process of differentiation), which you will encounter next. The rules here also underpin related rates problems, where you differentiate an equation that relates multiple changing quantities with respect to time. Beyond this course, derivatives are the foundation of differential equations, which model everything from population dynamics to electrical circuits.
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