Difficulty: Intermediate | Prerequisites: Sections 1 and 2 of this chapter (limit definition of the derivative). You need to know what a derivative is before learning the shortcuts.
Computing every derivative from scratch using the limit definition is slow and often brutal. This set of sections hands you the shortcut rules: power, sum, difference, constant multiple, product, quotient, and then the six trigonometric derivatives. Once you have these, most derivatives take seconds instead of a full page of algebra. You still need the limit definition (it is tested), but these rules handle the day-to-day work for the rest of the course.
The power rule (d/dx of x^n = nx^(n−1)) handles polynomials, roots, and reciprocals. The sum, difference, and constant-multiple rules let you break functions apart. The product and quotient rules handle multiplication and division of functions. The six trig derivatives must be memorised. Two special trig limits (sin x / x → 1 and (cos x − 1)/x → 0 as x → 0) underpin the proofs and appear in exam problems.
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 coefficient and reduce the exponent by one.
Constant rule
If f(x) = c (a constant), then f'(x) = 0. In simple terms, a flat horizontal line has zero slope.
Constant multiple rule
(d/dx)[c·f(x)] = c·f'(x). You can pull a constant factor out of the derivative.
Sum and difference rules
(d/dx)[f(x) ± g(x)] = f'(x) ± g'(x). Derivatives distribute over addition and subtraction.
Product rule
(d/dx)[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x). In simple terms, differentiate the first and leave the second alone, then leave the first alone and differentiate the second, then add.
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, over low squared."
Trigonometric derivatives
The derivatives of sin, cos, tan, sec, csc, and cot. The first two are foundational: (d/dx) sin x = cos x, and (d/dx) cos x = −sin x. The rest follow from the quotient rule.
Special trigonometric limits
lim(x→0) sin x / x = 1 and lim(x→0) (cos x − 1)/x = 0. These are used to prove the derivatives of sin and cos from the limit definition.
If f(x) = c, a constant, then f'(x) = 0. A horizontal line has no slope to speak of.
If f(x) = mx + b, a linear function, then f'(x) = m. The derivative of a line is its slope.
A useful trick: always simplify before differentiating. For instance, h(x) = (4x² − 1)/(2x + 1) factors as (2x+1)(2x−1)/(2x+1) = 2x − 1, so h'(x) = 2 by the linear rule.
For any real number n: if f(x) = x^n, then f'(x) = n·x^(n−1).
This works for positive integers, negative integers, fractions, and decimals. Rewrite roots and reciprocals as powers first:
√x = x^(1/2), so d/dx(√x) = (1/2)x^(−1/2) = 1/(2√x)
1/x = x^(−1), so d/dx(1/x) = −x^(−2) = −1/x²
³√x = x^(1/3), so d/dx(³√x) = (1/3)x^(−2/3)
1/x³ = x^(−3), so d/dx(1/x³) = −3x^(−4)
These rules let you break apart and reassemble:
(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)
Example: f(x) = 5/x − x^1.7 + 4. Rewrite as 5x^(−1) − x^1.7 + 4. Then f'(x) = −5x^(−2) − 1.7x^0.7.
(d/dx)[f·g] = f'·g + f·g'
Differentiate the first factor while leaving the second alone, then leave the first alone and differentiate the second. Add the two results.
Example: differentiate (x² − 3/x)(√x + x + 3). Set the first factor as f and the second as g, find f' and g' separately, then combine.
(d/dx)[f/g] = [f'·g − f·g'] / g²
The numerator follows the pattern: derivative of the top times the bottom, minus the top times the derivative of the bottom. The denominator is the bottom squared.
Example: find the tangent line to y = 2x/(x+1) at (1,1). Compute y' using the quotient rule: y' = [(2)(x+1) − (2x)(1)] / (x+1)² = 2/(x+1)². At x = 1, y' = 1/2. Tangent line: y − 1 = (1/2)(x − 1).
A horizontal tangent means f'(x) = 0. Set the derivative equal to zero and solve for x. For y = 2x³ + 3x² − 12x + 1, the derivative is y' = 6x² + 6x − 12 = 6(x+2)(x−1), which is zero at x = −2 and x = 1.
(d/dx) sin x = cos x
(d/dx) cos x = −sin x
(d/dx) tan x = sec²x
(d/dx) sec x = sec x · tan x
(d/dx) csc x = −csc x · cot x
(d/dx) cot x = −csc²x
The first two are proved from the limit definition using the special trig limits below. The derivatives of tan and sec follow from the quotient rule applied to sin/cos and 1/cos. Memorise at least sin, cos, tan, and sec.
lim(x→0) sin x / x = 1
lim(x→0) (cos x − 1) / x = 0
These are proved using the squeeze theorem and a geometric argument (areas of triangles and sectors on the unit circle). They are the engine behind the proof that (d/dx) sin x = cos x.
If lim(x→a) f(x) = 0, then lim(x→a) sin(f(x)) / f(x) = 1, and likewise lim(x→a) f(x) / sin(f(x)) = 1. This extends the basic limit to composite arguments.
Evaluating lim(x→0) sin(5x)/(3x): rewrite as (5/3) · sin(5x)/(5x). As x → 0, 5x → 0, so sin(5x)/(5x) → 1. The limit is 5/3.
Evaluating lim(x→0) sin(x² + 7x)/x: factor x from the argument, giving sin(x(x+7))/x. As x → 0, x(x+7) → 0, and sin(x(x+7))/[x(x+7)] → 1. So the limit becomes lim(x→0) (x+7) = 7.
The derivatives of sin x cycle with period 4: sin x → cos x → −sin x → −cos x → sin x. To find the 83rd derivative of sin x, compute 83 mod 4 = 3. The third derivative is −cos x, so the 83rd derivative of sin x is −cos x.
Rule | Formula |
|---|---|
Constant | d/dx(c) = 0 |
Linear | d/dx(mx + b) = m |
Power | d/dx(x^n) = n·x^(n−1) |
Constant multiple | d/dx(c·f) = c·f' |
Sum | d/dx(f + g) = f' + g' |
Difference | d/dx(f − g) = f' − g' |
Product | d/dx(f·g) = f'g + fg' |
Quotient | d/dx(f/g) = (f'g − fg') / g² |
sin x | cos x |
cos x | −sin x |
tan x | sec²x |
sec x | sec x · tan x |
csc x | −csc x · cot x |
cot x | −csc²x |
Special limit 1 | lim(x→0) sin x / x = 1 |
Special limit 2 | lim(x→0) (cos x − 1)/x = 0 |
The product rule appears whenever two quantities that both vary are multiplied together, for instance revenue = price × quantity, where both price and quantity change over time. The quotient rule handles ratios like concentration = amount/volume when both numerator and denominator change. Trigonometric derivatives are essential in any field involving oscillation: electrical engineering (AC circuits), mechanical engineering (vibrations), and physics (waves and pendulums).
Students often think the derivative of a product is the product of the derivatives: d/dx(f·g) = f'·g'. This is wrong. You must use the product rule.
Similarly, students assume d/dx(f/g) = f'/g'. The quotient rule has a subtraction in the numerator and a squared denominator. Get the order right: it is f'g − fg', not fg' − f'g.
When differentiating terms like 5/x, students sometimes try the quotient rule. It is far simpler to rewrite as 5x^(−1) and use the power rule.
Students frequently forget the negative sign in (d/dx) cos x = −sin x. A mnemonic: the "co" functions (cos, csc, cot) all pick up a negative sign in their derivatives.
⚠️ You will be expected to differentiate functions that combine powers, roots, trig, products, and quotients in a single expression. Practise mixed problems.
⚠️ Finding the tangent line using derivative rules (not the limit definition) is a standard exam question. Know the workflow: differentiate, evaluate at the point, write point-slope form.
⚠️ Trig limit problems (sin(5x)/3x style) appear on exams. Master the technique of rewriting so that sin(something)/something has matching arguments.
⚠️ The nth derivative of sin x (cyclic pattern mod 4) is a common exam question. Know the cycle: sin, cos, −sin, −cos.
Fill in the blank: d/dx(x^7) = ______
True or false: d/dx(f·g) = f'·g'
Fill in the blank: d/dx(sin x) = ______
True or false: lim(x→0) sin(x)/x = 0
What is the 100th derivative of cos x?
Q: Differentiate f(x) = x² + ³√x + 2x.
A: Rewrite as x² + x^(1/3) + 2x. Then f'(x) = 2x + (1/3)x^(−2/3) + 2.
Q: Find the equation of the tangent line to y = 2x/(x+1) at the point (1,1).
A: y' = 2/(x+1)² by the quotient rule. At x = 1, y' = 1/2. Tangent line: y − 1 = (1/2)(x − 1).
Q: Evaluate lim(t→0) t² / tan²(3t).
A: Rewrite tan(3t) = sin(3t)/cos(3t). Then t²/tan²(3t) = t²·cos²(3t)/sin²(3t) = [t/sin(3t)]²·cos²(3t). Since t/sin(3t) = (1/3)·(3t/sin(3t)) → 1/3, and cos(3t) → 1, the limit is (1/3)² = 1/9.
Q: Suppose f(π/3) = 4 and f'(π/3) = −2. Let g(x) = f(x)·sin x. Find g'(π/3).
A: g'(x) = f'(x)·sin x + f(x)·cos x. At x = π/3: g'(π/3) = (−2)(√3/2) + (4)(1/2) = −√3 + 2.
Q: Find all points where the tangent to y = 2x³ + 3x² − 12x + 1 is horizontal.
A: y' = 6x² + 6x − 12 = 6(x+2)(x−1) = 0 at x = −2 and x = 1. The points are (−2, 21) and (1, −6).
These differentiation formulas are the toolkit for everything that follows in the course. The chain rule (Section 5) extends these rules to composite functions. Implicit differentiation (Section 6) uses the same product and chain rules on equations not solved for y. Related rates (Section 8) are word problems built from implicit differentiation.
Related Terms / Search Tags
power rule, product rule, quotient rule, sum rule, difference rule, constant multiple rule, derivative of sin x, derivative of cos x, derivative of tan x, derivative of sec x, derivative of csc x, derivative of cot x, trig limits, sin x over x, horizontal tangent, differentiation formulas, MTH 132, Calculus I, Michigan State, Chapter 2