Differentials

Returning to the derivative $\dfrac{df}{dx}=\displaystyle\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=f'(x)$ (Leibniz's notation for the differential coefficient): is it meaningful to treat $\dfrac{df}{dx}$ as an actual quotient of two separate quantities $df$ and $dx$ — not merely a single symbol for a limit?

When $f$ is linear, $f(x)=mx+c$, the answer is unambiguously yes: $\Delta y = f(x+\Delta x)-f(x)=m\Delta x=f'(x)\Delta x$ exactly, for every $x$ and every $\Delta x$ (no limiting process needed, since a line's slope is the same everywhere). In this case

so $df/dx$ genuinely is the quotient of $df=\Delta f=\Delta y$ and $dx=\Delta x$.

Definition 8.4 (Differential). For a general differentiable $f:(a,b)\to\mathbb R$, $x\in(a,b)$, and $\Delta x$ an increment given to $x$, the differential of $f$ is

Taking $f(x)=x$ itself gives $dx=f'(x)\Delta x=1\cdot\Delta x=\Delta x$ — so the differential of the independent variable $x$ is simply its change, $dx=\Delta x$, and (8) can be rewritten $df=f'(x)\,dx$: precisely the "quotient" reading of $df/dx$ that motivated the whole discussion.

Geometric meaning. For $y=f(x)$, let $\Delta f=f(x+dx)-f(x)$ be the actual change in output along the curve, and let $dy$ (or $df$) be the change along the tangent line instead: since the tangent line has slope $f'(x)$, $dy=f'(x)\,dx$. From the picture, $\Delta f\approx dy=df=f'(x)dx$ — so $f'(x)$ can be viewed (approximately) as the quotient of $\Delta f$ and $\Delta x$, i.e. $df/dx$ is legitimately interpreted as a quotient of $df$ and $dx$, even though $f$ itself is nonlinear.

Differentials of standard functions (paired with their derivatives): $f(x)=x^n\Rightarrow df=nx^{n-1}dx$; $\ f(x)=\cos(x^2+7)\Rightarrow df=-\sin(x^2+7)(2x)\,dx$; $\ f(x)=\cot(x^2)\Rightarrow df=-\csc^2(x^2)(2x)\,dx$; $\ f(x)=\sin^{-1}x\Rightarrow df=\dfrac{dx}{\sqrt{1-x^2}}$; $\ f(x)=\tan^{-1}x\Rightarrow df=\dfrac{dx}{1+x^2}$; $\ f(x)=e^{3x^2-5x+7}\Rightarrow df=e^{3x^2-5x+7}(6x-5)\,dx$; $\ f(x)=\log(x^2+1)\Rightarrow df=\dfrac{2x}{x^2+1}dx$ — in every case, exactly the derivative multiplied by $dx$.

Properties of Differentials (for $f,g$ differentiable, $c$ a real constant):

1. If $f$ is constant, $df=0$.
2. If $f(x)=x$ (identity), $df=dx$.
3. $d(cf) = c f'(x)\,dx = c\,df$.
4. $d(f+g) = f'(x)dx+g'(x)dx = df+dg$.
5. Product rule: $d(fg) = f\,dg+g\,df$ (proved below).
6. Quotient rule: $d(f/g) = \dfrac{g\,df-f\,dg}{g^2}$, provided $g(x)\ne0$.
7. Chain rule: if $h=f\circ g$ is defined, $dh = f'(g(x))\,g'(x)\,dx$.
8. If $h(x)=e^{f(x)}$, $dh = e^{f(x)}f'(x)\,dx$.
9. If $f(x)>0$ and $g(x)=\log(f(x))$, $dg = \dfrac{f'(x)}{f(x)}\,dx$. …