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) h  0 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 x – 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 = -1 point: (1,-1) slope: -1 y – (-1) = (-1)(x – (1)) y + 1 = -x +1 y = -x is the tangent line to the original function at x = 1. Back to applications

Finding Relative Maxes or Mins The derivative is the easiest way to find the maximum or minimum value of a function. 1. Take the derivative 2. Set the derivative equal to 0 3. Solve for x 4. Take the derivative of the derivative (2 nd derivative) 5. Plug x values in 2 nd derivative If positive, minimum; if negative, maximum

Example Find the relative maxes and/or mins of y = x² - 4x dy/dx = 2x – x – 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 odd Find a web page that talks about derivatives in some way, write it down and a brief description of the page.

End of Show by Doug Robeson