Differentials and Partial Derivatives — Study Notes

Differentials and Partial Derivatives — Study Notes

This chapter takes the single most useful idea from calculus — that a smooth curve looks like a straight line up close — and turns it into a practical tool for estimating function values and tracking small errors. It then extends the derivative to functions of several variables through partial derivatives.

1. Linear approximation (one variable)

If \( f \) is differentiable at \( x_{0} \), the tangent line at that point hugs the graph closely for \(x\) near \( x_{0} \). This tangent line is the linear approximation:

\[ L(x) = f(x_{0}) + f'(x_{0})(x - x_{0}). \]

For \(x\) close to \( x_{0} \), \( f(x) \approx L(x) \). To estimate something like \( \sqrt[3]{27.2} \) by hand, pick a nearby “nice” point (\( x_{0} = 27 \)) where the function and its derivative are easy, then read off \( L \).

2. Differentials (one variable)

For \( y = f(x) \), with \( dx \) an independent increment in \(x\), the differential of \(y\) is

\[ dy = f'(x)\,dx. \]

The differential \( dy \) is the rise along the tangent line, whereas the true change \( \Delta y = f(x+\Delta x) - f(x) \) is the rise along the curve. For small \( \Delta x \) they nearly agree: \( \Delta y \approx dy \). This single relation drives every “approximate change” problem in the chapter.

3. Errors: absolute, relative, percentage

• Absolute error \( = \lvert \text{actual} - \text{approximate}\rvert \); it carries units.
• Relative error \( = \dfrac{\text{absolute error}}{\lvert\text{actual value}\rvert} \); it is unit-free.
• Percentage error \( = \text{relative error} \times 100 \).

A clean pattern worth memorising: differentiating a power-law quantity \( Q = k\,x^{n} \) gives \( \dfrac{dQ}{Q} = n\,\dfrac{dx}{x} \). So the percentage error multiplies by the exponent \(n\) (e.g. \( \times 2 \) for area, \( \times 3 \) for volume, \( \times \tfrac{1}{n} \) for an \(n\)th root).

4. Functions of several variables: limit and continuity

A function \( F(x,y) \) of two variables is visualised as a surface \( z = F(x,y) \). For a limit at \( (x_{0}, y_{0}) \) to exist, \( F(x,y) \) must approach the same value along every path to that point — not just along straight lines. The classic trap: if approaching along different lines \( y = mx \) gives answers that depend on \(m\), the limit does not exist, and \(F\) cannot be continuous there.

5. Partial derivatives

The partial derivative measures the rate of change in one direction while the other variables are frozen:

\[ \frac{\partial F}{\partial x}(x_{0},y_{0}) = \lim_{h\to 0}\frac{F(x_{0}+h,\,y_{0}) - F(x_{0},y_{0})}{h}. \]

In practice: to find \( \partial F/\partial x \), treat every other variable as a constant and differentiate normally. All the ordinary rules (product, quotient, chain, power) still apply. Notations: \( \dfrac{\partial F}{\partial x},\ F_{x},\ \partial_{x}F \).

6. Higher-order partials and Clairaut’s theorem

Differentiating twice gives second-order partials \( F_{xx}, F_{yy} \) and the mixed partials \( F_{xy}, F_{yx} \). Clairaut’s theorem: if \( F_{xy} \) and \( F_{yx} \) are continuous on the region, then \( F_{xy} = F_{yx} \) — the order of mixed differentiation does not matter. A function satisfying \( F_{xx} + F_{yy} = 0 \) is called harmonic (Laplace’s equation).

7. Linear approximation and differential (two and three variables)

For \( z = F(x,y) \) near \( (x_{0},y_{0}) \), the tangent plane plays the role of the tangent line:

\[ L(x,y) = F(x_{0},y_{0}) + F_{x}(x_{0},y_{0})(x - x_{0}) + F_{y}(x_{0},y_{0})(y - y_{0}), \]

and the total differential is \( dF = F_{x}\,dx + F_{y}\,dy \) (with the obvious \(z\)-term added for three variables).

8. Chain rule (function of a function)

If \( w = F(x,y) \) with \( x = x(t),\ y = y(t) \), then

\[ \frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt}. \]

If instead \( x = x(s,t),\ y = y(s,t) \), the same idea gives the partials \( \dfrac{\partial w}{\partial s} \) and \( \dfrac{\partial w}{\partial t} \), summing one product per intermediate variable. A tree diagram (\(w\) branching to \(x,y\), each branching to the parameters) keeps the bookkeeping straight.

9. Homogeneous functions and Euler’s theorem

\( F(x,y) \) is homogeneous of degree \(p\) if \( F(tx, ty) = t^{p} F(x,y) \) for all admissible \(t\). For such a function with continuous partials, Euler’s theorem states

\[ x\,\frac{\partial F}{\partial x} + y\,\frac{\partial F}{\partial y} = p\,F, \]

with the natural three-variable extension. A quick test for the degree: factor a common \( t^{p} \) out of \( F(tx,ty) \).

Common exam traps

• Confusing \( dy \) (tangent-line change) with \( \Delta y \) (actual change). The question's wording tells you which is wanted; “approximate change” means use the differential.
• Forgetting to convert a percentage increase into an actual increment \( dx \) before substituting (e.g. a \(1\%\) rise means \( dx = 0.01x \)).
• Treating \( x^{y} \) the same whether differentiating in \(x\) or in \(y\): power rule in \(x\) gives \( y x^{y-1} \); exponential rule in \(y\) gives \( x^{y}\log x \).
• Declaring a two-variable limit to exist after checking only one path. Path-dependence breaks the limit.
• Mixing up the plain partial sum \( F_{x} + F_{y} \) with Euler’s weighted sum \( xF_{x} + yF_{y} \).
• Dropping a sign in the quotient rule, or forgetting to square the denominator.