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If a differentiable map $F$ takes values in a level set of a differentiable scalar function $H$, the chain rule gives $D{H}_{F(x)}D{F}_x=0$. At a regular point, the image of the derivative lies in the kernel of the nonzero covector $DH$: the tangent space of the level set. For a map into the unit circle this forces rank at most one.

For an invertible matrix $A$ and any matrix $H$,Indeed $\det (A+tH)=\det (A)\det (I+t{A}^{-1}H)$, and multilinearity shows that the linear coefficient of $\det (I+tB)$ is $\mathrm{tr}B$. At a singular matrix the derivative is instead written $\mathrm{tr}(\mathrm{adj}(A)H)$; this polynomial formula holds for all $A$.

For positive real $a$, differentiating the Gamma function integral under the integral sign gives the displayed formula; successive derivatives insert higher powers of $\log t$. Dominated differentiation on compact positive-$a$ intervals is justified by exponential decay at infinity and integrability of $t^{a-1}|\log t{|}^n$ at zero. At $a=1$, these derivatives give moments used in logarithmic endpoint asymptotic expansions.

The usual zero-over-zero L'Hopital rule requires the denominator's derivative to be nonzero throughout a sufficiently small deleted neighborhood. A limit taken only where that derivative is nonzero is insufficient. An explicit counterexample uses $x_n=2^{-n}$, $h_n=4^{-n}$, $S(u)=3u^2-2u^3$ and $q(u)=16u^2(1-u{)}^2$. On $[3x_n/4,x_n]$ put $g=h_n$ and $f=h_{n}q((x-3x_n/4)/(x_n/4))$. On $[x_n/2,3x_n/4]$ put $g=h_{n+1}+(h_n-h_{n+1})S((x-x_n/2)/(x_n/4))$ and $f=0$. Extend $g=1,f=0$ beyond one. Values and first derivatives agree at the joins, both functions tend to zero, and $g$ is positive. Where $g^{\prime }\ne 0$, it lies inside a transition interval and $f^{\prime }=0$. Thus $f^{\prime }/g^{\prime }$ has natural-domain limit zero, but $f/g$ equals zero at $x_n$ and one at $7x_n/8$, so has no limit.

Suppose a degree-$N$ polynomial obeys $|{\partial }^{\alpha }Q(\xi )|\le C_{\alpha }|\xi {|}^{-\delta |\alpha |}|Q(\xi )|$ at large real frequency, with $\delta >0$. A nonzero constant derivative of total order $N$ gives $|Q|\gtrsim ⟨\xi {⟩}^{\delta N}$. Reciprocal differentiation then bounds ${\partial }^{\alpha }(1/Q)$ by $⟨\xi {⟩}^{-\delta N-\delta |\alpha |}$. The high-frequency reciprocal parametrix kernel therefore gains $\delta N$ Sobolev derivatives. Localize the distribution by a cutoff function equal to one near the target; the commutator lies outside that target and the reciprocal kernel makes its contribution smooth. This proves the displayed gain without demanding global regularity of the original distribution.

Here $C^{\infty }([0,1])$ means the real- or complex-valued functions continuous on $[0,1]$, smooth on $(0,1)$, with every interior derivative having a continuous extension to the endpoints; endpoint values of derivatives are these extensions. On this space of smooth functions on a compact interval, defineThese finite-valued seminorms increase with $k$ and separate points. In fact every $p_k$ is a norm, since it includes $p_0(f)=\Vert f{\Vert }_{\infty }$. The terminology seminorm includes norms; it does not assert that every measurement has a nontrivial kernel of a seminorm. Smooth convergence means $p_k(f_n-f)\to 0$ for every fixed $k$, equivalently uniform convergence of every derivative separately. It does not demand one tolerance or one convergence index that works for all orders simultaneously. The maximum convention is equivalent to the derivative supremum norm convention using a finite sum, since $p_k(f)\le {\sum }_{j=0}^k\Vert f^{(j)}{\Vert }_{\infty }\le (k+1)p_k(f)$.

For a worked example, $f(t)=t^3$ gives $p_0(f)=1$, $p_1(f)=3$, $p_2(f)=6$, and $p_k(f)=6$ for $k\ge 3$. The linear functional $\ell (f)=f^{\prime \prime }(1/2)$ obeys $|\ell (f)|\le p_2(f)$ and is continuous. Similarly, differentiation is a continuous linear operator $D:f\mapsto f^{\prime }$ for smooth convergence, because $p_k(Df)\le p_{k+1}(f)$.

The space of smooth functions on a compact interval is complete for this seminorm topology. A Cauchy sequence has uniform limits $g_j$ for its derivatives of each order. Passing to the limit in the fundamental theorem of calculus identitygives $g_j(t)-g_j(0)={\int }_0^t{g}_{j+1}(s)ds$, so $g_j^{\prime }=g_{j+1}$ and $g_0$ belongs to the same space. This is the completeness of the smooth-function metric argument; the metric there and the increasing-$p_k$ metric above induce the same smooth topology and the same Cauchy sequences.

A difference quotient divides a finite change in a function by the corresponding change in its argument. Its limit, when it exists, is the derivative.

Differentiability is the existence of a derivative at a point. For maps between finite-dimensional normed vector spaces, the existence of a Fréchet derivative requires a single linear map approximating the increment with error $o(\Vert h\Vert )$. Merely having partial derivatives need not provide this approximation; continuous partial derivatives on a neighbourhood do suffice.

Extend the displayed function by $f(0)=0$. Its difference quotient at zero is $h\sin (1/h)$, which tends to zero. Away from zero its derivative is $2x\sin (1/x)-\cos (1/x)$. At $x_k=1/(2\pi k)$ this derivative equals $-1$, whereas at $y_k=1/((2k+1)\pi )$ it equals $+1$. Thus differentiability implies continuity of the function but does not imply continuity of its derivative.

The function $f(x)=x^2\sin (1/x^2)$ for $x\ne 0$, extended by $f(0)=0$, is differentiable everywhere. At zero its derivative is zero by the difference quotient, while away from zero $f^{\prime }(x)=2x\sin (1/x^2)-2x^{-1}\cos (1/x^2)$ is unbounded near zero. Differentiability of a function is therefore weaker than continuity of its derivative.

If a real function has a finite derivative at $x$, its difference quotient tends to $f^{\prime }(x)$ and is bounded for sufficiently small nonzero increments. Multiplying it by the increment shows $f(x+h)-f(x)\to 0$, proving continuity at $x$. The converse fails: the absolute value function is continuous but not differentiable at zero.

A dominated statistical path is differentiable in quadratic mean at $P$ if its square-root probability density function has an L2 space derivative $(1/2)g\sqrt{p}$, with $g\in L^2(P)$. The score function is $g$. This formulation controls mass near zeros of $p$ as well as the derivative on the support of $P$; a pointwise log-density derivative is insufficient by itself.