Difficulty: Intermediate | Prerequisites: Limits, continuity, single-variable calculus basics (Calc I).
Derivatives measure how a function's output changes as its input changes. For single-variable functions this is a single number (the slope of the tangent line); for multivariable functions you take partial derivatives, one per input variable, each found by holding the other variables constant. These partial derivatives combine into a gradient vector and a tangent plane, giving you the tools to approximate, optimise, and analyse functions of several variables.
Derivative (single-variable)
The limit of the difference quotient, f'(x₀) = lim(h→0) [f(x₀+h) − f(x₀)] / h. It gives the instantaneous rate of change of f at x₀.
Think of it as: the slope of the graph at exactly one point, rather than between two points.
Difference quotient
The expression [f(x₀+h) − f(x₀)] / h, which measures the average rate of change of f over an interval of width h.
In simple terms, this is the "rise over run" between two nearby points on the curve. The derivative is what you get when you shrink that gap to zero.
Partial derivative
For a function of several variables, the derivative with respect to one variable while all other variables are held constant. Written as ∂f/∂xⱼ or f_xⱼ.
Think of it as: slicing through a surface along one axis and measuring the slope of that slice.
Tangent plane
The flat plane that best approximates a surface z = f(x, y) near a point (x₀, y₀). It is the multivariable analogue of the tangent line.
In simple terms, if you zoomed in very close to a smooth surface, it would look flat. The tangent plane is that flat sheet.
Linear approximation
Using the tangent plane (or tangent line, in one variable) to estimate the value of f near a known point. The formula is z ≈ f(x₀, y₀) + fₓ(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀).
Think of it as: a quick-and-dirty estimate that works well when you are close to the point, and poorly when you are far from it.
Gradient (∇f)
The vector of all partial derivatives: ∇f = (∂f/∂x₁, … , ∂f/∂xₙ). Points in the direction of steepest increase of f.
In simple terms, the gradient answers "which way is uphill, and how steep?"
Jacobian matrix
For a vector-valued function f: ℝⁿ → ℝᵐ, the m × n matrix whose (i, j) entry is ∂fᵢ/∂xⱼ. It generalises the gradient to functions with multiple outputs.
Think of it as: a table that collects every partial derivative of every output with respect to every input, all in one place.
The derivative f'(x₀) is the limit of the difference quotient as h → 0.
Geometrically, it is the slope of the tangent line to the graph of f at x₀.
The set of points where this limit exists is the domain of differentiability of f.
If f is differentiable at x₀, f is also continuous there (but the converse is not always true).
For f: ℝⁿ → ℝ, the partial derivative ∂f/∂xⱼ at a point x₀ is computed by holding every variable except xⱼ constant and differentiating as if f were a single-variable function of xⱼ.
The partial derivative measures the sensitivity of f to changes in xⱼ alone.
Existence of all partial derivatives at a point does not guarantee differentiability there. Example: f(x, y) = x^(1/3) y^(1/3) has both partial derivatives equal to 0 at the origin, yet is not differentiable at (0, 0) because the tangent plane z = 0 is a poor local approximation.
Two standard notations: subscript form f_xⱼ and Leibniz form ∂f/∂xⱼ.
To indicate evaluation at a specific point, write ∂f/∂xⱼ |_{x₀}.
Be careful to distinguish subscript notation (which names the variable) from index notation (which names the component). f_x means "partial with respect to x"; f₁ means "the first component of f" when f is vector-valued.
For f(x, y), the tangent plane at (x₀, y₀) is z ≈ f(x₀, y₀) + fₓ(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀).
This gives a local linear approximation: good near the point, increasingly inaccurate further away.
Worked example: for f(x, y) = e^(xy) ln(x² + y²), the tangent plane at (1, 1) is found by computing f(1, 1) = e ln 2, then finding ∂f/∂x and ∂f/∂y at (1, 1), and assembling L(x, y) = e ln 2 + fₓ(1,1)(x−1) + f_y(1,1)(y−1). This lets you estimate f(1.1, 1.2) without re-evaluating the full expression.
Single-variable derivative
f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}Partial derivative
\frac{\partial f}{\partial x_j}(\vec{x}_0) = \lim_{h \to 0} \frac{f(x_{10}, \ldots, x_{j0} + h, \ldots, x_{n0}) - f(\vec{x}_0)}{h}Tangent plane (two variables)
z \approx f(x_0, y_0) + \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0)Gradient vector
\nabla f(\vec{x}) = \left( \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right)Jacobian matrix
D f = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{bmatrix}Partial derivatives are how engineers compute how sensitive a design is to each of its parameters. If V = πr²h is the volume of a cylindrical tank, ∂V/∂r tells you how much volume you gain by widening the tank, and ∂V/∂h tells you how much you gain by making it taller. This kind of sensitivity analysis underpins everything from structural engineering to financial risk modelling.
Tangent-plane approximations are the mathematical backbone of linearisation in control systems: you approximate a complicated nonlinear system with a simpler linear one near an operating point.
Students often assume that if all partial derivatives exist at a point, the function must be differentiable there. This is false. The classic counterexample is f(x, y) = x^(1/3) y^(1/3), which has partials at the origin but is not differentiable there.
Confusing the notation f₁ (first component of a vector-valued function) with fₓ (partial with respect to x). These are different things when f maps ℝⁿ to ℝᵐ.
Treating the tangent plane as exact rather than an approximation. It is only accurate near the point of tangency; the error grows as you move away.
Forgetting to hold other variables constant when computing a partial derivative, and instead applying single-variable product or chain rules to the entire expression.
⚠️ You will almost certainly be asked to compute partial derivatives and a tangent plane for a given function at a given point. Practise the mechanics until they are automatic.
⚠️ A favourite exam question: give a function where partial derivatives exist at a point but the function is not differentiable there. Know the x^(1/3) y^(1/3) example cold.
⚠️ Notation matters. Be ready to switch between ∂f/∂x, fₓ, and Dⱼf without hesitation.
True or false: if f is differentiable at a point, then f is continuous there. (True)
True or false: if all partial derivatives of f exist at a point, then f is differentiable there. (False)
Fill in the blank: the tangent plane to z = f(x, y) at (x₀, y₀) is z ≈ f(x₀, y₀) + ____ + ____.
True or false: the gradient vector points in the direction of fastest decrease. (False; it points in the direction of fastest increase)
Fill in the blank: a partial derivative is computed by holding all variables except ____ constant.
Q: State the limit definition of the partial derivative ∂f/∂x for a function f(x, y) at the point (a, b).
A: ∂f/∂x (a, b) = lim(h→0) [f(a+h, b) − f(a, b)] / h.
Q: Find the tangent plane to f(x, y) = x² + y² at the point (1, 2).
A: f(1, 2) = 5. fₓ = 2x, so fₓ(1, 2) = 2. f_y = 2y, so f_y(1, 2) = 4. The tangent plane is z = 5 + 2(x − 1) + 4(y − 2).
Q: Give an example of a function whose partial derivatives both exist at a point but which is not differentiable there. Explain briefly why.
A: f(x, y) = x^(1/3) y^(1/3). At the origin, ∂f/∂x = 0 and ∂f/∂y = 0 (computed via the limit definition), but the tangent plane z = 0 does not approximate f well near (0, 0), so f is not differentiable there.
Q: What does the gradient vector ∇f tell you geometrically?
A: It points in the direction of the steepest increase of f, and its magnitude gives the rate of that increase. It is perpendicular to the level surfaces of f.
Q: Write down the Jacobian matrix for a function f: ℝ² → ℝ² where f(x, y) = (x² y, sin(xy)).
A: Df = [[2xy, x²], [y cos(xy), x cos(xy)]].
This material is the foundation for the chain rule in several variables, which multiplies Jacobian matrices to differentiate compositions. It also leads directly into directional derivatives and gradient-based optimisation (finding maxima and minima of functions of several variables).
The tangent plane concept generalises further to higher dimensions through the total derivative and the differential, which appear in the next set of notes.
Derivative, partial derivative, difference quotient, tangent line, tangent plane, linear approximation, linearisation, gradient, gradient vector, nabla, del operator, Jacobian, Jacobian matrix, total derivative, differential, rate of change, instantaneous rate of change, sensitivity, multivariable calculus, MATH 201, Calc III, vector calculus, differentiability, continuity, partial differentiation