Tangent line ($y=f(a)+f^{\prime }(a)(x-a)$)

The tangent line to a differentiable function at $a$ is its affine first-order approximation $f(a)+f^{\prime }(a)(x-a)$. For a differentiable concave function, it is an upper bound throughout a convex interval: the secant-slope inequality and passage to the derivative give $f(x)\le f(a)+f^{\prime }(a)(x-a)$. This upper bound constructs valid envelopes in adaptive rejection sampling.

A tangent line parallel to the vertical coordinate axis. For a regular level set $\Phi (x,y)=C$, the conditions $\partial \Phi /\partial y=0$ and $\partial \Phi /\partial x\ne 0$ make the tangent line vertical: the relation $d\Phi =0$ forces $dx=0$. The implicit function theorem then expresses $x$ locally as a differentiable function of $y$, although a graph $y(x)$ has no finite derivative there. The curve $x=y^2$ at the origin is a simple example.

A line tangent to two curves or to a curve at two distinct points. For functions $g_1,g_2$, contact at $x_1,x_2$ requires equal derivatives and equal intercepts: $g_1^{\prime }(x_1)=g_2^{\prime }(x_2)=m$ and $g_1(x_1)-m{x}_1=g_2(x_2)-m{x}_2$. The common-tangent construction uses the lower supporting line for phase coexistence.

For a nonzero real constant $a$, the tangent line to $y=e^{ax}$ at $x_{\ast }$ passes through the origin precisely when $e^{a{x}_{\ast }}-x_{\ast }a{e}^{a{x}_{\ast }}=0$. Thus $x_{\ast }=1/a$, $y_{\ast }=e$, and its slope is $k=ae$. This gives the unique such tangent line. If $a=0$, the graph is the horizontal line $y=1$ and no tangent line through the origin exists. The equality between the derivative and the slope from the origin, $f^{\prime }(x_{\ast })=f(x_{\ast })/x_{\ast }$, is useful for other differentiable functions too.

At a regular point of a differentiable plane curve, the normal line passes through that point perpendicular to its tangent line. If the tangent slope is a finite nonzero number $m$, the normal slope is $-1/m$, since the direction vectors $(1,m)$ and $(1,-1/m)$ have zero dot product. A horizontal tangent has a vertical normal, and a vertical tangent has a horizontal normal. For a regular level set $F(x,y)=c$, the nonzero gradient $\nabla F$ gives a normal direction because differentiating $F$ along the curve gives $\nabla F\cdot {\mathbf{r}}^{\prime }=0$.