Computations of Derivatives
• Thm- For any constant c,
• Note, when y = c, the slope of that line is always horizontal. Therefore, its derivative must equal 0
• Thm- Let f(x) = x, then
• Proof:
• Note: This means that the derivative of any linear function is equal to the coefficient
Power Rule• Let’s take a look at the different powers of x.
Can you see the pattern in the table?
F(x) F’(x)
1 0
X^1 1
X^2 2x
X^3 3x^2
X^4 4x^3
Power Rule cont’d
Power Rule - For any real number n,
Note: The power rule works for negative exponents, as well as fraction exponents.
General Derivative Rules
• Thm- If f(x) and g(x) are differentiable at x and c is any constant, then
• 1)
• 2)
• 3)
General Deriv. Rules• Remember, to rewrite any expressions so
they have exponents! And split the expression into separate terms!
• You try: Find the derivative of each:• 1)
• 2)
• 3)
HW Review p.139 #7-13 21-27, 33, 49
• 7) -32 33) 32s - 24• 9) 1/3 49) a) • 11) b)• 13) c)• 21)• 23)• 25)• 27)
Finding tangent line equations
• Now that we have the power rule, it is much easier to find the equations of tangent lines.
• EX: Find the tangent line to the graph at x = 2
Differentiability and Continuity
• If f is differentiable at x = c (the derivative is defined at c) then f is also continuous at c
Derivative Info
• The derivative can tell us when a function is increasing (+), decreasing (-), or horizontal (0)
• This makes finding the vertex of a function easier
Closure
• Journal Entry: How does the power rule work? What does a derivative tell us about a function?
• HW: p. 139 #15-19 all, 29 31 43 45
HW Review: 139 #15-19, 29 31 43 45
• 15) y = 32x - 48 45) A = f(x)• 16) y = -2/125 x + 3/25 B = h(x)• 17) y = -3x - 32 C = g(x)• 18) y = 1/12 x + 4/3• 19)a) 12e^x b) 25-8e^t c) e^(t-3)• 29) 0• 31) • 43) A - 3, B - 1, C - 2, D - 3
Derivative Rules so far:
• Power Rule• E^x• Constant Rule• Breaking up functions into individual terms
Closure
• Journal Entry: How does the graph of a function relate to the graph of its derivative?
• HW: Finish worksheet p.184 #5-23
HW Review p.184 5-23 odds
• 5) 3x^2 -2 21)• 7) 6x 23)• 9) 0• 11) • 13)• 15) -5x^(-3/2) -2• 17) • 19)