The General Power Rule

I.  Quiz

II.  Homework

III.  Investigation

Students will find the derivatives of the following:

1.  y = x^1/2  

2.  y = x^-1

3.  y = x^3/2

4.  y = x^-1/2

5.  y = x^-1/2

IV.  The General Power Rule

Theorem:  For any rational number m/n if u = x^m/n then u' = m/n x^{m/n - 1}

Proof for positive rational numbers:

Let y = uⁿ = x^m

then y' = mx^m-1

Also by the chain rule,

dy/dx = dy/du du/dx = nu^n-1 du/dx

so that

nu^n-1 du/dx = mx^m-1

and

du/dx = m/n x^m-1/u^n-1 = m/n x^m-1/x^m/n(n-1) = m/n x^m-1-m+m/n   

=m/n x^{m/n -1}

Example:

Find the derivative of the following function

y = sqrt(x² + 1)

We use the chain rule:

y = sqrt(u) = u^1/2

u = x² + 1

dy/du = 1/2 u^-1/2

du/dx = 2x

so that

dy/dx = 1/2 u^-1/2 (2x)

= x/sqrt(x² +1)

Exercises:  Find the following derivatives

A)  x sqrt(3-x)

B)  x²/sqrt(2x - 1)

C)  sqrt(1 - sqrt(1 - sqrt(1 - sqrt(x))))

V.  Proof of the Quotient Rule 

Proof of the Quotient Rule

(f(x)/g(x))' = (f(x)(g(x))^-1)' = (g(x))^-1f'(x) +  f(x)[-g'(x)/(g(x))²]   

=  (g(x)f'(x) - f(x)g'(x))/(g(x))²  

VI.  Discussion of the Project

We will discuss typical projects that you may want to try.