Higher Derivatives Example 1
The definition of higher derivatives and solving an example
List Comprehension and Patterns in Higher Derivatives in Python
Author: David Manuel
In this video, it is shown how to use Python to quickly find several derivatives of a function using Python's list comprehension tool. We can then find the patterns in the results allowing us to write a formula for all higher order derivatives of the function.
Transcript 14
Transcript 14
Problem: Find the first several derivatives of \(f(x)=\frac{1}{x}\) and generalize a pattern for the nth derivative.
Python Code:
from sympy import *
# List Comprehension
x=symbols('x')
f=1/x
# 1st 6 derivatives IN ONE LINE OF CODE!
fderivs=[diff(f,x,i) for i in range(1,7)]
#range(a,b)-> list of integers from a INCLUSIVE to b EXCLUSIVE
print(fderivs)
# exponent is 1 bigger than the derivative we took: x^(n+1)
# signs alternate (-1)^n or (-1)^(n+1). n=1 is negative, so (-1)^n
#factorial numbers in the numerator: n!
print('The formula for the nth derivative of f is (-1)^n*n!/x^(n+1)')
Open Python Notebook File
from sympy import *
# List Comprehension
x=symbols('x')
f=1/x
# 1st 6 derivatives IN ONE LINE OF CODE!
fderivs=[diff(f,x,i) for i in range(1,7)]
#range(a,b)-> list of integers from a INCLUSIVE to b EXCLUSIVE
print(fderivs)
# exponent is 1 bigger than the derivative we took: x^(n+1)
# signs alternate (-1)^n or (-1)^(n+1). n=1 is negative, so (-1)^n
#factorial numbers in the numerator: n!
print('The formula for the nth derivative of f is (-1)^n*n!/x^(n+1)')
Open Python Notebook File
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What the derivatives of a function tell us about the shape of its graph
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Optimization: MATH 171 Problems 1-3
Optimizing values in word problems
Antiderivatives: MATH 171 Problems 4-6
Proving facts about antiderivatives and a physics application
Riemann Sums and Definite Integrals: MATH 151 5.2 Problems 6-12
Using Reimann sums and the Fundamental Theorem of Calculus
Riemann Sums and Definite Integrals: MATH 151 5.3 Problems 6-12
Using Riemann sums and the Fundamental Theorem of Calculus
Fundamental Theorem of Calculus: MATH 171 Problems 1-3
Proving and then applying the Fundamental Theorem of Calculus
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
Integration by Parts: MATH 172 Problems 1 & 2
Proving the formula for integration by parts and a reduction formula
WIR2 20B M251 V8
Finding a vector equation for the tangent line to a three-dimensional vector function
WIR2 20B M251 V9
Finding the angle of intersection for two three-dimensional vector functions
WIR2 20B M251 V10
Finding the length of a three-dimensional curve
WIR2 20B M251 V11
Finding the curvature of a three-dimensional vector function
WIR4 20B M251 V9
Finding the absolute maximum and minimum values of a function of two variables on a rectangle
WIR4 20B M251 V10
Finding the absolute maximum and minimum values of a function of two variables on a triangular region
WIR8 20B M251 V1
Evaluating a line integral along half a circle
WIR8 20B M251 V2
Evaluating a line integral along a curve in space
WIR8 20B M251 V3
Evaluating a line integral along a curve
WIR8 20B M251 V4
Evaluating a line integral along a curve
WIR8 20B M251 V5
Evaluating a line integral along a curve in space
WIR8 20B M251 V6
Evaluating a line integral of a vector field along an curve
WIR8 20B M251 V7
Evaluating a line integral of a vector field along a curve in space
WIR8 20B M251 V8
Using a line integral to find the work done by a force field moving a particle along a curve
Matrix Derivative and Integral
Explaining how to find the derivative and integral of matrix and vector functions
MATH 152: Sequences Exercise 4
Finding the limit of a sequence
MATH 152: Sequences Exercise 6
Finding the limit of a sequence using logarithm properties
MATH 152: Sequences Exercise 7
Finding the limit of a sequence
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
MATH 152: Sequences Exercise 10
Determining if a sequence is increasing, decreasing, or non monotonic
WIR 20B M152 V45
Determining if a sequence converges or diverges
Partial Derivatives Exercise V4
Finding all higher order partial derivatives of a function with two variables
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
The Chain Rule (multivariable) Exercise V3
Using the Chain Rule for multivariable functions to find the rate a volume is changing
Finding the nth Taylor Polynomial Exercise
Finding the nth Taylor polynomial of the natural log function centered at 1
Approximation Using Taylor Polynomials Exercise
Using a Taylor polynomial to approximate the value of a function and estimating the error
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
WIR 20B M152 V51
Determining if a series converges or diverges
WIR 20B M152 V63
Using the Alternating Series Test to determine if a series converges or diverges
Using Variables in Python
Defining variables in Python and using the variables in equations
Introduction and Installation of Python
An introduction to installing and getting started with Python
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Plotting Expressions and Solving Equations Numerically in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Plotting Multiple Expressions in One Graph in Python
Graphing a piecewise function using Python
Numerical Calculations in Python
Performing basic numerical calculations in Python
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
Implicit Plots and Implicit Differentiation in Python
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
Definite and Indefinite Integrals in Python
Solving definite and indefinite integrals in Python
Finding Absolute (Global) Extrema in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
Approximating Definite Integrals with Riemann Sums in Python
Using Python to approximate a definite integral using left endpoint Riemann sums
List Comprehension and Numerical, Graphical, and Symbolic Limits in Python
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
For Loops and Recursive Sequences in Python
Using for loops in Python to generate a recursively defined sequence
Generating and Plotting Partials Sums of a Series in Python
Using Python to calculate the partial sums of a series to estimate the value of the series
Generating and Plotting Sequences in Python
Using Python to numerically, graphically, and analytically find the limit of a sequence
Areas Between Curves in Python
Using Python to identify the region between curves and then evaluate the area of the region
Calculating Work in Python: Hooke’s Law
Using Python to calculate the amount of work done in moving a spring using Hooke's Law
Calculating Work in Python: Pumping Liquid from a Tank
Using Python to calculate the work done in pumping the liquid from a tank whose ends are semicircles
Volumes of Revolution in Python
Finding the volume of a solid of revolution in Python
Radius and Interval of Convergence in Python
Using Python to find the radius and interval of convergence of a power series with the Ratio Test
Plotting a Function and Partial Sums of its Power Series in Python
Video 29 in a series of Python instructional videos
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions