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Differentiation

Learn the importance of differentiation in finding tangent lines, maximums and minimums, and the shape of curves in various fields of study such as physics, economics, and engineering. Explore definitions, basic rules, examples, and applications of derivatives.

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Differentiation

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  1. Differentiation Derivatives By: Doug Robeson

  2. 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

  3. 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

  4. A Tangent Line

  5. 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)

  6. 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

  7. More Examples • d((3x+4)/x²) = x²(3) – (3x+4)(2x) (x²)² = 3x² - 6x² - 8x x^4 = -3x – 8 x³

  8. Applications of Derivatives • Finding tangent lines • Finding relative maxes and mins

  9. 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: • Take derivative • Plug in x value (this gives you slope) • Put slope and point into point slope form of the equation of a line

  10. 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. Example Back to applications

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

  12. Example Find the relative maxes and/or mins of y = x² - 4x + 7. • dy/dx = 2x – 4 • 2x – 4 = 0 • x=2 • Second derivative (d²y/dx²) = 2 • Plugging anything into d²y/dx² and it’s positive, so x=2 is a relative minimum. Back to applications

  13. 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.

  14. End of Showby Doug Robeson

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