Numerical Methods

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Prerequisites

Algebra

Equation of a Line

The point-slope of the equation of a line with slope m and passing through the point (x0,y0) is yy0=m(xx0)

Calculus I

Derivatives

Differentiation Rules

General Formulas

Exponential and Logarithmic Formulas

Trigonometric Formulas

Inverse Trigonometric Formulas

Hyperbolic Formulas

Inverse Hyperbolic Formulas

Partial Derivatives

One of the conditions for the Existence and Uniqueness Theorem uses a first-order partial derivative of the dependent variable.

An ordinary differential equation (ODE) in the form dydx=f(x,y) states that the derivative of y with respect to x is a function dependent on both x and y, where y is a function of x. Since the function f is dependent on more than one variable, we will need to find partial derivatives, specifically the partial derivative of f with respect to y, i.e., fy or fy(x,y). But this is the similar to derivatives (without needing implicit differentiation since y is a dependent variable of f) in Calculus I if we treat all occurrences of x as either coefficients of y or constants.

Example 01

Find the partial derivative of f(x,y)=xy2x3y+x2 with respect to y.

The solution is f(x,y)y=x(2y)x3(1)+0=2xyx3.

Example 2

Find the partial derivative of f(x,y)=xy23tan(y)x+15ln(x) with respect to y.

The solution is f(x,y)y=x(23y13)sec2(y)x.

Practice Examples

Find the partial derivative of f(x,y) with respect to y. Verify your answers with a classmate or using software.

  1. f(x,y)=x2y24x5+sin2(x)ln(y)

  2. f(x,y)=xsin(y)ycos(x)

  3. f(x,y)=x2+y

  4. f(x,y)=xy

  5. f(x,y)=yx+4xy+e4yx2+1

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Last Modified: Monday, 31 August 2020 12:17 EDT