Derivative from the definition
The speedometer shows speed at one instant; a map shows the steepness of a path at one point. Both are derivatives.
The derivative of f at x is f′(x) = limh→0 [f(x + h) − f(x)]/h (when the limit exists). It gives the slope of the tangent and the rate of change. This definition is called the first principle.
Formula: f′(x) = lim h→0 [f(x + h) − f(x)] / h
Rule 1: First principle — Write f(x + h) − f(x), divide by h, then let h → 0.
- Easy: f(x) = x²: [(x + h)² − x²]/h = 2x + h, so f′(x) = 2x.
- Medium: f(x) = 3x + 2: [3h]/h = 3, so f′(x) = 3.
- Exam-type: f(x) = 5 (constant): f′(x) = 0.
Rule 2: Slope at a point — Put the point in f′(x).
- Easy: f(x) = x²: slope at x = 3 is 6.
- Medium: f(x) = x³: f′(x) = 3x², slope at x = 2 is 12.
- Exam-type: Position s = t² metres: speed at t = 4 s is 8 m/s.
The derivative is itself a function of x.
Dividing by h before expanding the bracket completely.
Derivative = limit of rise over run.
Practice questions
- 1Find f′(x) from first principle for f(x) = 4x.
- 2Find f′(x) from first principle for f(x) = x².
- 3Find the slope of y = x² at x = 5.
- 4Find the slope of y = x³ at x = 1.
- 5Find f′(x) for f(x) = 7.
- 6A stone falls s = 5t² metres. Find its speed at t = 2 s.
- 7Find the point on y = x² where the slope is 8.
- 8Find f′(2) for f(x) = x² + 3x.
Show answers (try them yourself first)
- 14
- 22x
- 310
- 43
- 50
- 610t = 20 m/s
- 72x = 8, x = 4: (4, 16)
- 8(2x + 3) at 2 = 7
Derivatives of powers and polynomials
Instead of using the definition every time, we use short rules.
For any real n: d/dx (xⁿ) = n xⁿ⁻¹. The derivative of a constant is 0 and d/dx (c f) = c f′. The derivative of a polynomial is found term by term.
Formula: d/dx (xⁿ) = n xⁿ⁻¹
Rule 1: Power rule — The power comes down, the power drops by one.
- Easy: d/dx (x⁵) = 5x⁴.
- Medium: d/dx (x⁻²) = −2x⁻³.
- Exam-type: d/dx (√x) = (1/2) x^(−1/2).
Rule 2: Polynomials — Differentiate each term.
- Easy: d/dx (3x⁴ − 2x² + 7x − 5) = 12x³ − 4x + 7.
- Medium: d/dx (x³ + 6x) = 3x² + 6.
- Exam-type: d/dx (5x − 9) = 5.
The derivative of x is 1; of a constant is 0.
Differentiating a constant term as if it contained x.
Bring the power down, reduce it by one.
Practice questions
- 1Differentiate x⁷.
- 2Differentiate 5x³ − 2x.
- 3Differentiate 2x⁴ + 3x² + 9.
- 4Differentiate 1/x.
- 5Differentiate x³ − 3x² + 2x − 1 and find f′(1).
- 6Find the slope of y = x⁴ − 2x at x = 1.
- 7Find the points on y = x³ − 3x where the slope is 0.
- 8Differentiate 3√x.
Show answers (try them yourself first)
- 17x⁶
- 215x² − 2
- 38x³ + 6x
- 4−1/x²
- 53x² − 6x + 2, f′(1) = −1
- 64 − 2 = 2
- 73x² − 3 = 0, x = ±1: (1, −2) and (−1, 2)
- 8(3/2) x^(−1/2)
Sum, product and quotient rules
Complicated functions are built from simple ones. Rules tell how to differentiate the combinations.
If u and v are differentiable: (u ± v)′ = u′ ± v′; (uv)′ = u′v + uv′ (product rule); (u/v)′ = (u′v − uv′)/v² (quotient rule, v ≠ 0).
Formula: (uv)′ = u′v + uv′
Rule 1: Product rule — Differentiate the first, keep the second, then the other way round.
- Easy: d/dx [x²(x + 1)] = 2x(x + 1) + x² = 3x² + 2x.
- Medium: d/dx [(x + 2)(x − 3)] = (x − 3) + (x + 2) = 2x − 1.
- Exam-type: d/dx [x³(2x − 5)] = 3x²(2x − 5) + 2x³ = 8x³ − 15x².
Rule 2: Quotient rule — (Top′ × Bottom − Top × Bottom′) ÷ Bottom².
- Easy: d/dx [x/(x + 1)] = 1/(x + 1)².
- Medium: d/dx [(x² )/(x − 1)] = (2x(x − 1) − x²)/(x − 1)² = (x² − 2x)/(x − 1)².
- Exam-type: d/dx [1/x²] = −2/x³.
In the quotient rule the numerator order matters: (u′v − uv′).
Writing (uv)′ = u′v′.
Quotient: Low d-High minus High d-Low, over low squared.
Practice questions
- 1Differentiate (x + 1)(x + 2).
- 2Differentiate x²(x − 1).
- 3Differentiate (x + 1)/(x − 1).
- 4Differentiate x/(x² + 1).
- 5Find f′(2) for f(x) = x²(x + 1).
- 6Find f′(1) for f(x) = (x² + 1)/(x + 1).
- 7Find the slope of y = 2x(x² − 3) at x = 1.
- 8Differentiate (2x + 1)(3x − 2).
Show answers (try them yourself first)
- 12x + 3
- 23x² − 2x
- 3−2/(x − 1)²
- 4(1 − x²)/(x² + 1)²
- 5f′ = 3x² + 2x: 12 + 4 = 16
- 6(2x(x + 1) − (x² + 1))/(x + 1)² = (4 − 2)/4 = 1/2
- 70 (y′ = 6x² − 6)
- 812x − 1
Derivatives of trigonometric functions
The sine wave goes up and down; its slope is a cosine wave, and the slope of the cosine wave is minus a sine wave.
d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x; d/dx (cot x) = −cosec² x; d/dx (sec x) = sec x tan x; d/dx (cosec x) = −cosec x cot x. (x in radians.)
Formula: d/dx (sin x) = cos x
Rule 1: Direct — Use the table.
- Easy: d/dx (3 sin x) = 3 cos x.
- Medium: d/dx (sin x + cos x) = cos x − sin x.
- Exam-type: d/dx (tan x) = sec²x.
Rule 2: With product and quotient rules — Combine the rules.
- Easy: d/dx (x sin x) = sin x + x cos x.
- Medium: d/dx (sin x / x) = (x cos x − sin x)/x².
- Exam-type: d/dx (sin x cos x) = cos²x − sin²x.
The derivatives of the "co" functions (cos, cot, cosec) carry a minus sign.
Forgetting the minus sign in d/dx (cos x).
Co-functions go negative.
Practice questions
- 1Differentiate 2 sin x − 3 cos x.
- 2Differentiate x² sin x.
- 3Differentiate tan x − x.
- 4Find f′(0) for f(x) = sin x + cos x.
- 5Differentiate cos x / x.
- 6Differentiate sin x/(1 + cos x).
- 7Find the slope of y = sin x at x = π/3.
- 8Differentiate x sec x.
Show answers (try them yourself first)
- 12 cos x + 3 sin x
- 22x sin x + x² cos x
- 3sec²x − 1
- 41
- 5(−x sin x − cos x)/x²
- 61/(1 + cos x)
- 7cos π/3 = 1/2
- 8sec x + x sec x tan x