I Want to Plot Second Derivative of Inverse Secant Function with SymPy

[Freya_the_Goddess]

Hi all,

I have the formula for the first derivative for secant inverse:
D_{x} \sec^{-1} \ x = \frac{1}{|x| \sqrt{x^{2}-1}}

with |x| > 1

I can manually find the second derivative (the second is very complex). Then copy the equation manually, but it can’t be plotted. Thus, I comment the last 2 lines. How to plot the second derivative of \sec^{-1} \ x ?

using Plots, LaTeXStrings, SymPy, Plots.PlotMeasures
gr()

function pitick(start, stop, denom; mode=:text)
    a = Int(cld(start, π/denom))
    b = Int(fld(stop, π/denom))
    tick = range(a*π/denom, b*π/denom; step=π/denom)
    ticklabel = piticklabel.((a:b) .// denom, Val(mode))
    tick, ticklabel
end

function piticklabel(x::Rational, ::Val{:text})
    iszero(x) && return "0"
    S = x < 0 ? "-" : ""
    n, d = abs(numerator(x)), denominator(x)
    N = n == 1 ? "" : repr(n)
    d == 1 && return S * N * "π"
    S * N * "π/" * repr(d)
end

function piticklabel(x::Rational, ::Val{:latex})
    iszero(x) && return L"0"
    S = x < 0 ? "-" : ""
    n, d = abs(numerator(x)), denominator(x)
    N = n == 1 ? "" : repr(n)
    d == 1 && return L"%$S%$N\pi"
    L"%$S\frac{%$N\pi}{%$d}"
end

a, b = 0, π

f(x) = sec.(x)
xs = range(a, b, length=150)
ys = f.(xs)

plot(ys, xs, color=:red, ytick=pitick(a, b, 2; mode=:latex), 
	xlims=(-3,3), ylims=(0, π), framestyle=:zerolines,
	linestyle=:solid,  linecolor=:red2,
	legend=:topleft, label=L"\sec^{-1} \ x", 
	bottom_margin=3mm,
	size=(800, 400), tickfontsize=10)# the inverse function

@syms x::(real,positive)
h(x) = 1/(abs(x)*sqrt(x^2-1))
xs1 = range(1, b, length=150)
ys1 = h.(xs1)

# diff(h(x),x)

plot!(xs1, ys1, label=L"D_{x} \sec^{-1} \ x = \frac{1}{|x| \sqrt{x^{2} - 1}}", framestyle=:zerolines)

#h2(x) = -x/((x^2 - 1)^(3/2)*abs(x)) - (re(x)*Derivative(re(x), x) + im(x)*Derivative(im(x), x))*sign(x)/(x*sqrt(x^2 - 1)*Abs(x)^2)
#plot!(h2, label=L"D^{2}_{x} \sec^{-1} \ x ", framestyle=:zerolines)

[Sevi]

This doesn’t directly answer your question, but maybe there is still some problem with the formulas. WolframAlpha claims that the first derivative should have x^2 instead of |x| in the denominator, for example.

EDIT: My bad, I didn’t realize the two expressions are the same anyway. I’ll leave the link to WA here just in case someone else find it useful.

arcsec'(x) - Wolfram|Alpha

You can also compare the three different functions by putting them into the prompt with commas arcsec(x), arcsec'(x), arcsec''(x)

[rocco_sprmnt21]

try using CalculusWithJulia

using Plots, LaTeXStrings, CalculusWithJulia
gr()

y(x)=asec(x)

Dy(x)=y'(x)

DDy(x)=Dy'(x)

xs1 = range(1, 4, length=150)
ys1 = Dy.(xs1)

plot!(xs1, ys1, label=L"D_{x} \sec^{-1} \ x = \frac{1}{|x| \sqrt{x^{2} - 1}}")


ys2 = DDy.(xs1)

plot(xs1,ys2, label=L"D^{2}_{x} \sec^{-1} \ x ", framestyle=:zerolines)

[j_verzani]

Does the plot recipe not just work here:

using SymPy, Plots
@syms x
plot(diff(asec(x), x, 2), 0, pi, ylim = (-10, 10))

[Freya_the_Goddess]

This is from Calculus book (Purcell):

Which one is correct for the D_{x} \sec^{-1} \ x ?

[Freya_the_Goddess]

I am trying your method by using CalculusWithJulia and asec, but there is a lack of \sec^{-1} plot at the x < -1 and I need to add another plot fro x<-1 and one for x>1, this is the old code :

using Plots, LaTeXStrings, Plots.PlotMeasures
gr()

function pitick(start, stop, denom; mode=:text)
    a = Int(cld(start, π/denom))
    b = Int(fld(stop, π/denom))
    tick = range(a*π/denom, b*π/denom; step=π/denom)
    ticklabel = piticklabel.((a:b) .// denom, Val(mode))
    tick, ticklabel
end

function piticklabel(x::Rational, ::Val{:text})
    iszero(x) && return "0"
    S = x < 0 ? "-" : ""
    n, d = abs(numerator(x)), denominator(x)
    N = n == 1 ? "" : repr(n)
    d == 1 && return S * N * "π"
    S * N * "π/" * repr(d)
end

function piticklabel(x::Rational, ::Val{:latex})
    iszero(x) && return L"0"
    S = x < 0 ? "-" : ""
    n, d = abs(numerator(x)), denominator(x)
    N = n == 1 ? "" : repr(n)
    d == 1 && return L"%$S%$N\pi"
    L"%$S\frac{%$N\pi}{%$d}"
end

a, b = 0, π

f(x) = sec.(x)
xs = range(a, b, length=150)
ys = f.(xs)

plot(ys, xs, color=:red, ytick=pitick(-b, b, 2; mode=:latex), 
	xlims=(-3,3), ylims=(-π/2,π),framestyle=:zerolines,
	linestyle=:solid,  linecolor=:red2,
	legend=:topright, label=L"\sec^{-1} \ x", 
	bottom_margin=3mm,
	size=(800, 400), tickfontsize=10)# the inverse function

h(x) = 1/(abs(x)*sqrt(x^2 - 1))
h2(x) = 2x/((1+x^2)^2)
xs1 = range(1, b, length=150)
ys1 = h.(xs1)

# diff(h(x),x)

plot!(ys1, xs1, label=L"D_{x} \sec^{-1} \ x = \frac{1}{|x| \sqrt{x^{2}-1}}", framestyle=:zerolines)
#plot!(h2, label=L"D^{2}_{x} \cot^{-1} \ x = \frac{2x}{(1 + x^{2})^{2}}", framestyle=:zerolines)

\sec^{-1} at the x< -1 is plotted here:

With code modified from yours:

using Plots, LaTeXStrings, Plots.PlotMeasures, CalculusWithJulia
gr()

function pitick(start, stop, denom; mode=:text)
    a = Int(cld(start, π/denom))
    b = Int(fld(stop, π/denom))
    tick = range(a*π/denom, b*π/denom; step=π/denom)
    ticklabel = piticklabel.((a:b) .// denom, Val(mode))
    tick, ticklabel
end

function piticklabel(x::Rational, ::Val{:text})
    iszero(x) && return "0"
    S = x < 0 ? "-" : ""
    n, d = abs(numerator(x)), denominator(x)
    N = n == 1 ? "" : repr(n)
    d == 1 && return S * N * "π"
    S * N * "π/" * repr(d)
end

function piticklabel(x::Rational, ::Val{:latex})
    iszero(x) && return L"0"
    S = x < 0 ? "-" : ""
    n, d = abs(numerator(x)), denominator(x)
    N = n == 1 ? "" : repr(n)
    d == 1 && return L"%$S%$N\pi"
    L"%$S\frac{%$N\pi}{%$d}"
end

a, b = 1, 2π

y(x) = asec(x)
Dy(x)=y'(x)
DDy(x)=Dy'(x)

xs = range(-10, -1, length=150)
xs1 = range(a, b, length=150)
ys1 = Dy.(xs1)
ys2 = DDy.(xs1)

plot(xs, y, color=:red, ytick=pitick(-b, b, 2; mode=:latex), 
	xlims=(-3,3), ylims=(-π/2,π),framestyle=:zerolines,
	linestyle=:solid,  linecolor=:red2,
	legend=:bottomleft, label=L"\sec^{-1} \ x", 
	bottom_margin=3mm,
	size=(800, 460), tickfontsize=10)# the inverse function
plot!(xs1, y, color=:red, label="")

plot!(xs1, ys1, label=L"D_{x} \sec^{-1} \ x = \frac{1}{|x| \sqrt{x^{2}-1}}",
	color=:blue2)
plot!(xs, ys1, label="",color=:blue2)

plot!(xs1, ys2, label=L"D^{2}_{x} \sec^{-1} \ x", color=:green3)
plot!(xs, ys2, label="", color=:green3)

The great thing is it can plot D_{x}^{2} \sec^{-1} :

it must be imaginary → the horizontal line for the first and second derivative at x<-1
thanks!

[Sevi]

Sorry, the formula is correct as you wrote it, I didn’t see that the term in the square root is also different (so both versions are equivalent).

I edited my original answer.