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Derivatives
Shrink a gap to zero and find the slope at a point — then point uphill in many dimensions.
This one’s part of How Models Learn Pro — every lesson, every interactive, and new ones as they land. About 12 minutes.
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Inside “Derivatives”
What you’ll learn
- Average rate of change
- Shrink the gap
- The derivative is a function
- Rules for common functions
- Two inputs: partial derivatives
- The gradient points uphill
Key terms
- Secant
- A straight line through two points on a curve; its slope is the average rate of change between them.
- Tangent
- The straight line through a point on a curve with the same slope as the curve there; close to that point it hugs the curve.
- Derivative
- The slope of a function at each point: the limit of (f(x + h) − f(x)) / h as h → 0, written f′(x).
- Partial derivative
- How fast a function of several inputs changes when only one input moves and the rest are held fixed, written ∂f/∂x.
- Gradient
- The vector of all partial derivatives, ∇f; where it isn’t zero, it points in the direction of steepest increase.
- Contour line
- A line on a map joining points where a function has the same value.
- Saddle point
- A flat point (zero gradient) that is uphill in some directions and downhill in others, so it is neither a peak nor a dip.
Sources
- OpenStax Calculus Volume 1, 3.1 Defining the Derivative — Difference quotient (f(a + h) − f(a))/h as a secant slope; tangent slope and derivative as its limit as h → 0; instantaneous rate of change.
- OpenStax Calculus Volume 1, 3.2 The Derivative as a Function — The derivative function f′(x) = lim (f(x + h) − f(x))/h as h → 0 (eq. 3.9).
- OpenStax Calculus Volume 1, 3.3 Differentiation Rules — Power rule d/dx xⁿ = nxⁿ⁻¹ (Theorem 3.3, positive integers n; the lesson only uses n = 2 and 3).
- OpenStax Calculus Volume 1, 3.5 Derivatives of Trigonometric Functions — d/dx sin x = cos x (Theorem 3.8, eq. 3.11).
- OpenStax Calculus Volume 1, 3.9 Derivatives of Exponential and Logarithmic Functions — d/dx eˣ = eˣ (Theorem 3.14); d/dx ln x = 1/x for x > 0 (Theorem 3.15).
- Goodfellow, Bengio and Courville, Deep Learning, Chapter 3: Probability and Information Theory — Logistic sigmoid σ(x) = 1/(1 + exp(−x)) (eq. 3.30) and its derivative σ(x)(1 − σ(x)) (eq. 3.34).
- OpenStax Calculus Volume 3, 4.3 Partial Derivatives — Partial derivative as a limit with the other variable held fixed (eqs. 4.12–4.13).
- OpenStax Calculus Volume 3, 4.6 Directional Derivatives and the Gradient — Gradient ∇f = f_x i + f_y j; the rate of change is greatest along ∇f with value ‖∇f‖ (Theorem 4.13); ∇f is normal to level curves when non-zero (Theorem 4.14).
- Goodfellow, Bengio and Courville, Deep Learning, Chapter 4: Numerical Computation — §4.3: the gradient is the vector of partial derivatives; it points directly uphill and the negative gradient directly downhill; gradient descent x′ = x − ε∇f (eq. 4.5).
- 3Blue1Brown, Essence of Calculus: The paradox of the derivative — The derivative as the value the ratio approaches as the step shrinks towards 0; secants approaching the tangent.