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
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