Differentiation

By: Doug Robeson

What is differentiation for?

Finding the slope of a tangent line

Finding maximums and minimums

Finding the shape of a curve

Finding rates of change and average rates of change

Physics, Economics, Engineering, and many other areas of study

Definitions of the Derivative

The slope of a tangent line to a function

Change in y over the change in x: dy/dx

The limit definition lim f(x + h) - f(x)

h0 h

A Tangent Line

Basic Derivative Rules

Power Rule: d(x^n) = nx^(n-1)

Constant Rule: d(c) = 0

Note: u and v are functions

Product Rule: d(uv)=uv’ + vu’

Quotient Rule: d(u/v)=vu’ - uv’ v²

Chain Rule: d(f(g(x)))= f’(g(x))*g’(x)

Examples of Derivatives

d(x^3) = 3x^2

d(4) = 0

d((x+5)(3x-4)) = (x+5)(3) + (3x-4)(1) = 3x +15 + 3x – 4 = 6x + 11

d((2x² + 4)³) = 3(2x² + 4)²(4x) = 12x(2x² + 4)²

More Examples

d((3x+4)/x²) = x²(3) – (3x+4)(2x) (x²)² = 3x² - 6x² - 8x x^4 = -3x – 8 x³

Applications of Derivatives

Finding tangent lines

Finding relative maxes and mins

Finding Tangent Lines

• The derivative is the equation for finding tangent slopes to a function

• To find the tangent line to a function at a point:

1. Take derivative

2. Plug in x value (this gives you slope)

3. Put slope and point into point slope form of the equation of a line

Example Find the tangent line to y = 3x³ + 5x² - 9 when

x = 1.dy/dx = 9x² + 10x

slope = 9(1)² + 10(1) = 9 – 10 = -1

Have slope, need point: y = 3(1)³ + 5(1)² - 9 = -1point: (1,-1) slope: -1y – (-1) = (-1)(x – (1))y + 1 = -x +1y = -x is the tangent line to the original function at x = 1.

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Finding Relative Maxes or Mins

The derivative is the easiest way to find the maximum or minimum value of a function.

1. Take the derivative2. Set the derivative equal to 03. Solve for x4. Take the derivative of the derivative (2nd

derivative)5. Plug x values in 2nd derivative

If positive, minimum; if negative, maximum

Example

Find the relative maxes and/or mins of y = x² - 4x + 7.

1. dy/dx = 2x – 4

2. 2x – 4 = 0

3. x=2

4. Second derivative (d²y/dx²) = 2

5. Plugging anything into d²y/dx² and it’s positive, so x=2 is a relative minimum.

Back to applications

For more practice with Derivatives

Homework: Page 125, 1-53 oddFind a web page that talks about

derivatives in some way, write it down and a brief description of the page.

End of Showby Doug Robeson