04 // Interactive Formula Compendium // CALCULUS-ALGEBRAd/dx [f(g(x))] = f'(g(x)) · g'(x)
Chain Rule, Product Rule & Quotient Rule — Differentiation Sheet
The Chain Rule states that the derivative of a composite function h(x) = f(g(x)) is the outer derivative evaluated at the inner function multiplied by the inner derivative: h'(x) = f'(g(x))·g'(x) (or dy/dx = dy/du · du/dx).
Formula Reference
Main Formula
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Rearranged Forms
d/dx [u · v] = u'v + u v' (Product Rule)
d/dx [u / v] = (u'v − u v') / v² (Quotient Rule, v ≠ 0)
d/dx [xⁿ] = n · xⁿ⁻¹ (Power Rule)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| h'(x) | Composite Power Derivative d/dx [(ax + b)ⁿ] | dimensionless |
| a | Inner Slope (a = g'(x)) | 1 |
| b | Inner Intercept (b) | 1 |
| x | Evaluation Point (x) | 1 |
| n | Outer Power Exponent (n) | 1 |
Inputs & Parameters
[1]
[1]
[1]
[1]
Calculated Result (Composite Power Derivative d/dx [(ax + b)ⁿ])
h'(x) =4,116dimensionless
Formula Notes & Background
- For a small increment Δx, the inner function changes by Δu = g(x + Δx) − g(x) ≈ g'(x)Δx.
- The outer function changes by Δy = f(u + Δu) − f(u) ≈ f'(u)Δu.
- Dividing by Δx and taking lim_{Δx→0} yields dy/dx = lim (Δy/Δu)(Δu/Δx) = f'(g(x)) · g'(x).
Step-by-Step Examples
#1: Differentiate h(x) = (3x + 1)⁴ and evaluate h'(2).
Outer f(u) = u⁴ ⇒ f'(u) = 4u³Inner u = 3x + 1 ⇒ u' = 3x = 2
h'(x) = 4(3x + 1)³ × 3 = 12(3x + 1)³; at x = 2: 12 × 7³ = 12 × 343
→ h'(2) = 4,116
#2: Differentiate y = x² / (x + 1) using the Quotient Rule at x = 1.
u = x², u' = 2xv = x + 1, v' = 1
y' = [2x(x + 1) − x²(1)] / (x + 1)² = (x² + 2x) / (x + 1)²; at x = 1: 3 / 4
→ y'(1) = 0.75
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Frequently Asked Questions
Why is the Chain Rule the engine behind neural network backpropagation?+
A deep neural network is a nested composition of layer activations L(f_K(...f_1(x))); backpropagation applies the Chain Rule from output to input to compute gradients ∂L/∂w_i.
How do you remember the sign order in the Quotient Rule?+
Use 'Low d-High minus High d-Low over Low-Squared': (v·u' − u·v') / v², starting with the denominator v times the derivative of the numerator u'.
Can the Quotient Rule be derived from the Product and Chain Rules?+
Yes: write u/v = u · v⁻¹ and apply the Product Rule: u'v⁻¹ + u(−1·v⁻²·v') = (u'v − uv') / v².