Differentials and Partial Derivatives: One Mark Questions, Book Back Answers and Notes

About Differentials and Partial Derivatives

Medium ~150 min study

Suppose the side of a metal cube expands slightly when heated. How much does its volume change? You do not need to recompute the volume exactly: the derivative gives a very good estimate. That is the main idea of the first half of this unit. Linear approximation replaces a curve by its tangent line near a known point, and the differential $dy = f'(x)\,dx$ gives the approximate change in $y$.

The second half moves to functions of several variables, such as the volume of a cylinder, which depends on both radius and height. A partial derivative measures how the function changes when one variable changes and all the others are held fixed. The same approximation ideas carry over: the linear approximation and the total differential of a function of two variables use both partial derivatives.

For the exam, practise two kinds of question. Numerical ones ask you to estimate a value like $\sqrt[3]{26}$ or to find absolute, relative and percentage error. Proof questions ask you to verify that the mixed partial derivatives are equal, to check whether a function is harmonic, or to verify Euler's theorem for a homogeneous function. These ideas are used again in later courses on differential equations and physics.

What you'll learn

• Find the linear approximation of a function of one or two variables at a given point.
• Estimate values such as square roots and cube roots, and changes in area or volume, without a calculator.
• Compute absolute, relative and percentage errors of an approximation.
• Find first and second order partial derivatives, and verify that mixed partial derivatives are equal.
• Test whether a function is homogeneous and find its degree.
• Verify and apply Euler's theorem for homogeneous functions.

Before you start

• Rules of differentiation for functions of one variable.
• Limits and continuity of real-valued functions.
• Area and volume formulas for common shapes such as the sphere, cube and cylinder.

Differentials and Partial Derivatives explained

Differentials and Partial Derivatives: Main Ideas

Linear Approximation

Near a point $x_0$, a differentiable function is close to its tangent line. The linear approximation is $L(x) = f(x_0) + f'(x_0)(x - x_0)$. Choose $x_0$ close to the required value and such that $f(x_0)$ and $f'(x_0)$ are easy to find. For example, to estimate $\sqrt{36.1}$ take $f(x) = \sqrt{x}$ and $x_0 = 36$. The estimate is good only when $x - x_0$ is small.

Errors in Approximation

Absolute error is the absolute difference between the actual value and the approximate value. Relative error is the absolute error divided by the actual value. Percentage error is the relative error multiplied by 100. Relative and percentage errors let you compare accuracy fairly: an error of 1 cm matters far more in measuring a pencil than in measuring a road.

Differentials

For $y = f(x)$, the differential is $dy = f'(x)\,dx$, where $dx$ is a small change in $x$. The actual change $\Delta y$ is measured along the curve, while $dy$ is measured along the tangent line, and the two are close when $dx$ is small. A derivative is a rate; a differential is an amount of change. The usual rules carry over, for instance $d(uv) = u\,dv + v\,du$.

Partial Derivatives

For a function $u = f(x, y)$, the partial derivative with respect to $x$ is found by treating $y$ as a constant and differentiating with respect to $x$. Similarly for $y$. The textbook begins with limits and continuity for functions of two variables, then defines partial derivatives as limits. Geometrically, $\frac{\partial u}{\partial x}$ is the slope of the surface in the direction of the x-axis. For a function of two variables, the linear approximation at $(x_0, y_0)$ and the total differential $du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy$ both use the two partial derivatives together.

Second Order Partial Derivatives and Harmonic Functions

Differentiating a partial derivative again gives second order partial derivatives. The mixed ones, $\frac{\partial^2 u}{\partial x \partial y}$ and $\frac{\partial^2 u}{\partial y \partial x}$, are equal whenever they are continuous. This result is Clairaut's theorem. A function $u(x, y)$ is called harmonic if it satisfies Laplace's equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$. Note that only the pure second derivatives are added, not the mixed ones.

Homogeneous Functions and Euler's Theorem

A function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$ for every $\lambda > 0$. For example, $x^3 + 3xy^2$ is homogeneous of degree 3. Euler's theorem states that for such a function $x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf$. This often saves a long calculation: once you know the degree, the combination of partial derivatives is immediate.

Common mistakes to avoid

• Forgetting to treat the other variables as constants during partial differentiation, for example differentiating y in xy when finding the partial derivative with respect to x.
• Choosing a point far from the required value for linear approximation, which gives a poor estimate.
• Treating the derivative (a rate) and the differential (an approximate change) as the same thing.
• Using Euler's theorem on a function without first checking that it is homogeneous and finding its degree.
• Adding the mixed partial derivatives while testing for a harmonic function; Laplace's equation uses only the pure second derivatives.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the difference between a derivative and a differential?

The derivative f′(x) is the rate of change of the function at a point. The differential dy = f′(x) dx is the approximate change in the function when x changes by a small amount dx. One is a rate, the other is an amount.

How do I choose the point x₀ for a linear approximation?

Pick x₀ close to the value you want and such that the function and its derivative are easy to compute there. To estimate the square root of 36.1, choose x₀ = 36, since its square root is exactly 6. A point further away gives a less accurate answer.

Why do we need partial derivatives?

Many quantities depend on more than one variable. The volume of a cylinder depends on both radius and height. A partial derivative tells you how the quantity changes when only one of these is changed and the rest are kept fixed.

What makes a function harmonic?

A function u(x, y) is harmonic if it satisfies Laplace's equation, that is, the second partial derivative with respect to x plus the second partial derivative with respect to y equals zero. Such functions appear in physics, for example in steady-state heat flow and electrostatics.

Can the order of partial differentiation be switched?

Yes, when the mixed partial derivatives are continuous in the region, they are equal. This is Clairaut's theorem. Polynomials, exponentials and the usual trigonometric functions satisfy this condition wherever they are defined, so for them you may differentiate in whichever order is easier.

When is a function homogeneous of degree n?

When replacing x by λx and y by λy (λ > 0) multiplies the whole function by λⁿ. For a polynomial, this means every term has the same total degree n. Only then can you apply Euler's theorem.