MTH2010 Chap.3 Partial Derivatives, Tangent Planes and Linear Approximation
Partial Derivatives, Tangent Planes and Linear Approximation
Partial derivatives measure how a function of several variables changes when only one input moves. Formally, the partial derivative with respect to x at a point is the limit of a difference quotient in which y is held fixed, and in practice it is computed by treating the other variables as constants.
The limit definition remains essential at points where the usual rules do not apply, such as the boundary between two pieces of a piecewise function, and the mid-semester papers test exactly that situation. Differentiating again produces four second-order partial derivatives.
The seminar slides fix an order convention for mixed partials and state Clairaut's theorem, which guarantees the two mixed partials agree when they are continuous near the point, along with a counterexample where they do not. The second theme is approximation.
The tangent plane to the graph at a point is spanned by the tangent vectors of the two coordinate slices, and its equation defines the linearisation L, which approximates the function nearby. A function is differentiable at a point when the error f - L, divided by the distance to the point, tends to zero.
Having both partial derivatives is not enough for this, but continuous partial derivatives near the point are, and differentiability in turn implies continuity. The week closes with partial differential equations such as the wave, heat and Laplace equations, whose solutions are checked by computing the required partial derivatives.
What this chapter covers
- 01
Partial derivatives: hold the other variables fixed and differentiate
- 02
Limit definition: required at seams of piecewise functions
- 03
Higher partials: the order convention for mixed partials and Clairaut's theorem
- 04
Tangent plane: built from the two slice tangent vectors and a normal
- 05
Linearisation: value plus partials times displacements, used for estimates
- 06
Differentiability: relative error tends to zero; continuous partials suffice
- 07
Differentiable implies continuous, and its contrapositive as a quick test
- 08
Partial differential equations: verifying wave, heat and Laplace solutions
Worked example · free
Linear estimate of an exponential surface
- 1Value at the point: f(2, 0) = 2 e^0 = 2.
- 1Partials: f_x = e^y = 1 and f_y = x e^y = 2 at (2, 0).
- 1Linearisation: L(x, y) = 2 + (x - 2) + 2y.
- 1Estimate: L(2.1, -0.05) = 2 + 0.1 - 0.1 = 2.0, against the true value 2.1 e^(-0.05), about 1.998.
Key terms
- Mixed partial derivative
- A second-order derivative taken once with respect to each of two different variables.
- Clairaut's theorem
- A result stating that the two mixed partial derivatives agree at a point when they are continuous on a disc around it.
- Tangent plane
- The plane through a point of a surface that contains the tangent vectors of all smooth curves on the surface through that point.
- Differentiable function
- A function whose linearisation error, divided by the distance to the point, tends to zero.
- Relative error
- The size of the linearisation error divided by the distance from the base point.
- Laplace equation
- A partial differential equation stating that the sum of the pure second partial derivatives of a function is zero.
Partial Derivatives, Tangent Planes and Linear Approximation FAQ
When must I use the limit definition of a partial derivative?
Use it where the derivative rules do not apply, for example at the join between two pieces of a piecewise function or where a fractional power is not differentiable. Write the difference quotient explicitly and evaluate the limit, including one-sided limits if needed.
Does having partial derivatives mean a function is differentiable?
No. Partial derivatives can exist at a point where the function is not differentiable and even where it is not continuous. Continuous partial derivatives on a disc around the point do guarantee differentiability.
How do I write the tangent plane to a graph?
Evaluate the function and its two partial derivatives at the point, then write z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b). Check that the point itself satisfies the equation.
What is the order convention for mixed partials in this unit?
The seminar slides read the notation from the right: the variable nearest the function is differentiated first. When continuity holds, Clairaut's theorem makes the order irrelevant, but the convention matters for counterexamples.
How do I verify a solution of a partial differential equation?
Compute each partial derivative that appears in the equation, substitute them, and simplify until both sides agree. For compositions such as F(x - ct), the chain rule supplies the derivatives.
Exam move
Drill partial derivatives until routine, then spend most revision time on the two places marks are lost: definition-based partials at seams and the justification step of linear approximation. For seams, write the difference quotient in full and evaluate one-sided limits separately.
For approximations, always produce four lines in the same order: the value, the two partials, the linearisation, and the estimate, followed by the differentiability reason. Rework one tangent-plane problem with a product or quotient so the partials are not trivial, then verify a Laplace solution as practice for the partial differential equation questions.
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