Calculus

Calculus studies how quantities change. In machine learning, calculus appears whenever a loss changes with parameters, inputs, time, noise level, or coordinates.

For a scalar function $f:\mathbb{R}\to \mathbb{R}$, the derivative is:

$$\frac{df}{dx}=\underset{ϵ\to 0}{\lim }\frac{f(x+ϵ)-f(x)}{ϵ}$$

For a multivariate function $f:{\mathbb{R}}^d\to \mathbb{R}$, the gradient is:

$${\nabla }_{x}f={\left[\frac{\partial f}{\partial x_1},\dots ,\frac{\partial f}{\partial x_d}\right]}^{\top }$$

The chain rule is the core of backpropagation:

$$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))g^{\prime }(x)$$

See Chain rule for composition, vector-Jacobian products, and backpropagation notation.

Jacobian View

For a vector-valued function $f:{\mathbb{R}}^d\to {\mathbb{R}}^k$, the derivative is the Jacobian:

$$J_f(x)=\frac{\partial f}{\partial x}\in {\mathbb{R}}^{k\times d},(J_f{)}_{ij}=\frac{\partial f_i}{\partial x_j}$$

For a composition $h(x)=f(g(x))$:

$$J_h(x)=J_f(g(x))J_g(x)$$

Backpropagation usually avoids materializing the full Jacobian. It propagates vector-Jacobian products:

$$v^{\top }{J}_f(x)$$

from the output loss back toward earlier variables.

Directional Derivative

The directional derivative of $f$ at $x$ in direction $u$ is:

$$D_{u}f(x)=\underset{ϵ\to 0}{\lim }\frac{f(x+ϵu)-f(x)}{ϵ}={\nabla }_{x}f(x{)}^{\top }u$$

This explains why the gradient gives the steepest local increase under an ${\ell }_2$ direction constraint:

$$\underset{\Vert u{\Vert }_2=1}{\max }{\nabla }_{x}f(x{)}^{\top }u=\Vert {\nabla }_{x}f(x){\Vert }_2$$

The maximizing direction is $u={\nabla }_{x}f(x)/\Vert {\nabla }_{x}f(x){\Vert }_2$ when the gradient is nonzero.

Second-order information describes curvature. For a scalar loss $\mathcal{L}$, this is captured by the Hessian:

$$H_{\mathcal{L}}(x)={\nabla }_x^2\mathcal{L}(x)$$

See Jacobian and Hessian for vector-valued derivatives, curvature, and Jacobian/Hessian products.

For a small perturbation $\Delta x$, a second-order approximation is:

$$f(x+\Delta x)\approx f(x)+\nabla f(x{)}^{\top }\Delta x+\frac{1}{2}\Delta x^{\top }{H}_f(x)\Delta x$$

This is the local picture behind curvature, Newton-style updates, sharpness analysis, and some stability arguments.

Gradient Descent Connection

For parameters $\theta$ and loss $\mathcal{L}(\theta )$, the simplest update is:

$${\theta }_{t+1}={\theta }_t-\eta {\nabla }_{\theta }\mathcal{L}({\theta }_t)$$

where $\eta$ is the learning rate. In stochastic training, the full gradient is replaced by an estimate from a mini-batch:

$$g_t=\frac{1}{B}\sum _{i\in {\mathcal{B}}_t}{\nabla }_{\theta }\mathcal{L}(f_{\theta }(x_i),y_i)$$

The math note defines the derivative object. Optimizer behavior belongs in Optimization and Optimizer.

Key Ideas

• Derivatives describe local sensitivity.
• Gradients point in the direction of steepest local increase for scalar functions.
• Optimization usually moves against the gradient of a loss.
• Chain rules compose derivatives across layers, time steps, or computational graphs.
• Continuous-time models, diffusion, flow matching, and dynamics use derivatives with respect to time or noise level.

Practical Checks

• Is the function scalar-valued or vector-valued?
• Are derivatives taken with respect to inputs, parameters, time, or coordinates?
• Is the gradient used for optimization, sensitivity analysis, or dynamics?
• Does the implementation compute exact gradients, approximate finite differences, or stopped gradients?
• Is the code using gradients, Jacobian-vector products, vector-Jacobian products, or Hessian-vector products?
• Does the notation distinguish population loss, mini-batch loss, and single-example loss?

Related

• Matrix calculus
• Chain rule
• Jacobian and Hessian
• Vector norm and similarity
• Gradient descent
• Backpropagation
• Optimization
• Optimizer
• Stochastic gradient
• Flow matching