## DIFFERENTITATION OF PARAMETRIC FUNCTION

Differentiation of parametric function is another interesting method in the topic differentiation.

In this method we will have two functions known as x and y. Each function will be defined using another third variable.

Example 1 :

Find  dy/dx for the below parametric functions.

x = a cos θ and y = b sin θ

Solution :

Here the two variables x and y are defined using third variable θ.

By using the function x we can only find dx/dθ and by using the function y we can only find dy/dθ. By dividing these two derivatives we can find dy/dx.

x = a cos θ                       y = b sin θ

dx/dθ = a (-sin θ)             dy/dθ = b cos θ

dx/dθ = - a sin θ

Now we can find dy/dx

dy/dx = (dy/dθ)/(dx/dθ)

dy/dx = (b cos θ)/(- a sin θ)

dy/dx = -b/a (cos θ/ sin θ)

dy/dx = -(b/a) cot θ

Example 2 :

Find dy/dx by using the parametric functions

x = a sec3 θ  and  y = b tan3 θ

Solution :

Here the two variables x and y are defined using third variable θ.

By using the function x we can only find dx/dθ and by using the function y we can only find dy/dθ.

By dividing these two derivatives we can find dy/dx.

x = a sec3 θ

dx/dθ  =  3a sec² θ(sec θ tan θ)

=  3a sec³ θ tan θ

y  =  b tan³ θ

dy/dθ  =  3b tan² θ  sec² θ

Now we have divide those two derivatives to find dy/dx

dy/dx = (dy/dθ)/(dx/dθ)

=  (3b tan² θ  sec² θ)/(3a sec³ θ tan θ)

=  (b/a) (tan θ/sec θ)

=  (b/a) (sin θ/cos θ)/(1/cos θ)

=  (b/a) (sin θ/cos θ) x (cos θ/1)

=  (b/a) sin θ

Question 1 :

Differentiate the following

x = a cos3t, y = a sin3t

Solution :

x = a cos3t

dx/dt  =  a(3cos2t)(-sint)

dx/dt  =  -3a cos2t sint----(1)

y = a sin3t

dy/dt  =  a(3sin2t)(cost)

=  3a sin2t cost ----(2)

Divide (2) by (1)

dy/dx  =  (3asin2t cost) / (-3acos2t sint)

=  -sint/cost

dy/dx  =  -tant

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here

Kindly mail your feedback to v4formath@gmail.com

WORD PROBLEMS

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and Venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

## Recent Articles

1. ### Circles Worksheet

May 19, 22 12:17 PM

Circles Worksheet

2. ### SAT Math Practice Worksheets

May 19, 22 12:14 PM

SAT Math Practice Worksheets - Topic wise worksheet with step by step explanation for each question