Derivative Rules

Active Calculus (CC BY-SA) · activecalculus.org

Five rules make differentiation mechanical: power, product, quotient, chain, and implicit. Once you know them, you never need the limit definition again for standard functions.

Power rule

d/dx [xⁿ] = n xⁿ⁻¹. This handles every polynomial. Combined with linearity (derivative of a sum is the sum of derivatives, constants pull out), it covers all polynomials instantly.

Scheme

; Power rule: d/dx[x^n] = n * x^(n-1)
; Numerical derivative for verification
(define (deriv f)
  (let ((h 0.00001))
    (lambda (x) (/ (- (f (+ x h)) (f (- x h))) (* 2 h)))))

; f(x) = x^3, f'(x) = 3x^2
(define (f x) (* x x x))
(define (f-prime x) (* 3 x x))  ; power rule

(define f-numerical (deriv f))

(display "f'(2) power rule: ") (display (f-prime 2)) (newline)    ; 12
(display "f'(2) numerical:  ") (display (/ (round (* (f-numerical 2) 1000)) 1000)) (newline)

; g(x) = 5x^4 - 3x^2 + 7, g'(x) = 20x^3 - 6x
(define (g x) (+ (- (* 5 (expt x 4)) (* 3 x x)) 7))
(define (g-prime x) (- (* 20 (expt x 3)) (* 6 x)))

(define g-numerical (deriv g))

(display "g'(1) power rule: ") (display (g-prime 1)) (newline)    ; 14
(display "g'(1) numerical:  ") (display (/ (round (* (g-numerical 1) 1000)) 1000))

Product rule

d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x). The derivative of a product is not the product of the derivatives. Both functions contribute to the rate of change.

Scheme

; Product rule: (f*g)' = f'g + fg'
(define (deriv f)
  (let ((h 0.00001))
    (lambda (x) (/ (- (f (+ x h)) (f (- x h))) (* 2 h)))))

; f(x) = x^2, g(x) = sin(x)
; (f*g)' = 2x*sin(x) + x^2*cos(x)
(define (f x) (* x x))
(define (g x) (sin x))
(define (product x) (* (f x) (g x)))

(define (product-prime x)
  (+ (* (* 2 x) (sin x)) (* (* x x) (cos x))))

(define product-numerical (deriv product))

(display "At x=1:") (newline)
(display "Product rule: ") (display (/ (round (* (product-prime 1) 10000)) 10000)) (newline)
(display "Numerical:    ") (display (/ (round (* (product-numerical 1) 10000)) 10000))

Chain rule

d/dx [f(g(x))] = f'(g(x)) · g'(x). The derivative of a composition is the product of the derivatives, each evaluated at the right point. This is the most used rule in practice.

Scheme

; Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
(define (deriv f)
  (let ((h 0.00001))
    (lambda (x) (/ (- (f (+ x h)) (f (- x h))) (* 2 h)))))

; h(x) = sin(x^2)
; h'(x) = cos(x^2) * 2x  (chain rule)
(define (h x) (sin (* x x)))

(define (h-prime x) (* (cos (* x x)) (* 2 x)))

(define h-numerical (deriv h))

(display "At x=1:") (newline)
(display "Chain rule: ") (display (/ (round (* (h-prime 1) 10000)) 10000)) (newline)
(display "Numerical:  ") (display (/ (round (* (h-numerical 1) 10000)) 10000)) (newline)

(display "At x=2:") (newline)
(display "Chain rule: ") (display (/ (round (* (h-prime 2) 10000)) 10000)) (newline)
(display "Numerical:  ") (display (/ (round (* (h-numerical 2) 10000)) 10000))

Quotient rule

d/dx [f(x)/g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]². Low d-high minus high d-low, over the square of what's below.

Scheme

; Quotient rule: (f/g)' = (f'g - fg') / g^2
(define (deriv f)
  (let ((h 0.00001))
    (lambda (x) (/ (- (f (+ x h)) (f (- x h))) (* 2 h)))))

; q(x) = sin(x) / x
; q'(x) = (cos(x)*x - sin(x)*1) / x^2
(define (q x) (/ (sin x) x))

(define (q-prime x)
  (/ (- (* (cos x) x) (sin x)) (* x x)))

(define q-numerical (deriv q))

(display "At x=1:") (newline)
(display "Quotient rule: ") (display (/ (round (* (q-prime 1) 10000)) 10000)) (newline)
(display "Numerical:     ") (display (/ (round (* (q-numerical 1) 10000)) 10000))

Implicit differentiation

When y is defined implicitly by an equation like x² + y² = 1, differentiate both sides with respect to x, treating y as a function of x. Then solve for dy/dx. For the circle: 2x + 2y(dy/dx) = 0, so dy/dx = -x/y.

Scheme

; Implicit differentiation: x^2 + y^2 = 1 (unit circle)
; dy/dx = -x/y

; At the point (3/5, 4/5) on the circle:
(define x 0.6)  ; 3/5
(define y 0.8)  ; 4/5

(display "x^2 + y^2 = ") (display (+ (* x x) (* y y))) (newline) ; 1.0
(display "dy/dx = -x/y = ") (display (/ (- x) y)) (newline) ; -0.75

; Verify numerically: y = sqrt(1 - x^2)
(define (y-explicit x) (sqrt (- 1 (* x x))))

(define h 0.00001)
(define numerical-slope
  (/ (- (y-explicit (+ x h)) (y-explicit (- x h))) (* 2 h)))

(display "Numerical dy/dx = ") (display (/ (round (* numerical-slope 10000)) 10000))

Notation reference

Rule	Formula	When to use
Power	nxⁿ⁻¹	x raised to a constant
Product	f'g + fg'	Two functions multiplied
Quotient	(f'g - fg')/g²	One function over another
Chain	f'(g(x)) · g'(x)	Function of a function
Implicit	differentiate both sides, solve for dy/dx	y defined by an equation

Neighbors

∫ Ch.4 The Derivative — what these rules are shortcuts for
∫ Ch.6 Applications — using derivatives to optimize and sketch curves
∫ Ch.2 Trigonometry — trig derivatives: (sin x)' = cos x, (cos x)' = -sin x
🍞 Capucci 2021 — the chain rule as lens composition: backprop is the backward pass of a categorical optic
🤖 ML Ch.3 — the chain rule is the core of backpropagation in neural networks
∞ Real Analysis Ch.5 — the mean value theorem is the rigorous foundation for derivative rules

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