In this tutorial, we discuss how to solve the problems from the topics like limits, derivatives, and Taylor's series expansion using SymPy.

# Load required libraries
import sympy as smp 
from sympy import *

x, y = smp.symbols('x y')

f = x**2+y
f

$\displaystyle x^{2} + y$

smp.sin(x)

$\displaystyle \sin{\left(x \right)}$

smp.sec(x)

$\displaystyle \sec{\left(x \right)}$

other basic math function

• smp.log(x)
• smp.log(x,2)

x**(smp.Rational(3,2))

$\displaystyle x^{\frac{3}{2}}$

Computing Limits

SymPy can compute symbolic limits with the limit function. We use the following syntax

• To evaluate the limit $$\lim_{x\rightarrow x_0}f(x)$$ we use smp.limit(f(x),x,x0)
• To compute the right hand limit $$\lim_{x\rightarrow x_0^+}f(x)$$ we use smp.limit(f(x),x,x0,dir="+")
• To compute the left hand limit $$\lim_{x\rightarrow x_0^-}f(x),$$ we use smp.limit(f(x),x,x0,dir="-")

Compute the limits of the following: $$\lim_{x\rightarrow \pi} \sin (x/2 +\sin(x))$$

smp.limit(smp.sin(x/2+smp.sin(x)),x,smp.pi)

$\displaystyle 1$

smp.Limit(smp.sin(x/2+smp.sin(x)),x,smp.pi)

$\displaystyle \lim_{x \to \pi^+} \sin{\left(\frac{x}{2} + \sin{\left(x \right)} \right)}$

$$\lim_{x\rightarrow 0^+}{2e^{1/x}\over e^{1/x}+1} $$

smp.limit(2*smp.exp(1/x) / (smp.exp(1/x) + 1),x,0,dir='+') # one sided limit

$\displaystyle 2$

$$\lim_{x\rightarrow\infty}{\cos x-1\over x}$$

smp.limit((smp.cos(x)-1)/x,x,smp.oo)

$\displaystyle 0$

Exercise:

1. Find $\displaystyle \lim_{x\rightarrow 0}\Big({\sqrt{1-\cos 2x}\over\sqrt{2}x}\Big)$
2. Find $\displaystyle\lim_{x\rightarrow 0}x^4\sin({\pi\over 4})$

Derivatives

$${d\over dx}\Big({1+\sin x\over 1-\cos x}\Big)^2$$

smp.diff(((1+smp.sin(x))/(1-smp.sin(x)))**2,x)

$\displaystyle \frac{2 \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}} + \frac{2 \left(\sin{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{3}}$

smp.diff(((1+smp.sin(x))/(1-smp.cos(x)))**2,x)

$\displaystyle \frac{2 \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} - \frac{2 \left(\sin{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{3}}$

$${d\over dx}(\log_5(x))^{x/2}$$

smp.diff(smp.log(x,5)**(x/2),x)

$\displaystyle \left(\frac{\log{\left(x \right)}}{\log{\left(5 \right)}}\right)^{\frac{x}{2}} \left(\frac{\log{\left(\frac{\log{\left(x \right)}}{\log{\left(5 \right)}} \right)}}{2} + \frac{1}{2 \log{\left(x \right)}}\right)$

$${d\over dx}f(x+g(x))$$

f, g = smp.symbols('f g', cls=smp.Function)
g = g(x)
f = f(x+g)

smp.diff(f,x)

$\displaystyle \left(\frac{d}{d x} g{\left(x \right)} + 1\right) \left. \frac{d}{d \xi_{1}} f{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}=x + g{\left(x \right)} }}$

Higher Derivatives

# First derivatives
smp.diff(smp.exp(x**4),x)

$\displaystyle 4 x^{3} e^{x^{4}}$

$${d^2\over dx^2}e^{x^4}$$

# Second derivative
smp.diff(smp.exp(x**4),x,x)

$\displaystyle 4 x^{2} \left(4 x^{4} + 3\right) e^{x^{4}}$

$${\partial^7\over \partial x\partial y^2\partial z^4}e^{xyz}$$

z = smp.symbols('z')

expr = smp.exp(x*y*z) 
diff(expr, x,y,y,z,z,z,z)

$\displaystyle x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$

## Equivalently
diff(expr,x,y,2,z,4)

$\displaystyle x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$

## Alternative
deriv = smp.Derivative(expr,x,y,2,z,4)
deriv

$\displaystyle \frac{\partial^{7}}{\partial z^{4}\partial y^{2}\partial x} e^{x y z}$

deriv.doit()

$\displaystyle x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$

Exercise:

1. Find the 5th derivative of $(2x+5)$
2. Compute the derivative of $\log(4x^2-1)$
3. Compute the derivative of ${x+3\over (x-1)(x+2)}$
4. Compute the derivative of $x^2\log3x$

Taylor's Series:

Any smooth function $f(x)$ can be approximated by using series expansion. The Taylor's series of the function $f(x)$ at $x=x_0$ is $$ f(x)=f(x_0) + {(x-x_0)f'(x_0)\over 1!}+ {(x-x_0)^2f"(x_0)\over 2!}+\cdots+\infty $$ The series expansion at $x_0=0$ is known as Maclaurin's series.

Series Expansion

SymPy can compute asymptotic series expansions of functions around a point. To compute the expansion of $f(x)$ around the point $x=x_0$ terms of order $x^n.$ The following sytax will give the series expansion.

  f(x).series(x,x0,n)

By default x0=0 and n=6

fun = smp.sin(x)
fun.series(x,0,8)

$\displaystyle x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \frac{x^{7}}{5040} + O\left(x^{8}\right)$

All $x$ terms with power greater than or equal to $x^8$ are omitted.

Taylor's series of the function $e^x$ about $x=1.$

smp.exp(x).series(x,x0=6,n=4)

$\displaystyle e^{6} + \left(x - 6\right) e^{6} + \frac{\left(x - 6\right)^{2} e^{6}}{2} + \frac{\left(x - 6\right)^{3} e^{6}}{6} + O\left(\left(x - 6\right)^{4}; x\rightarrow 6\right)$

Exercise:

1. Expand $log(1+\sin^2x)$ in powers of $x$ as far as the term in $x^6.$
2. Expand $\log_e x$ about $x=1.$
3. Expand $e^{x\sin x}$ about $x=0.$