Introduction to derivatives

Reading

• Sections 3.1

Practice problems

• Section 3.1: 3, 5, 11, 13, 15, 21, 33, 35, 37, 41
• To turn in: 3.1 4, 26, 34, 38

Notes

Definition of Derivative

• The derivative is a formal description of the concept of “instantaneous rate of change” that we looked at when we discussed limits.
• Geometrically can be thought of as the slope of the tangent line, which is a limit of the slopes of the secant lines.

• Definition:

  The derivative of \(f(x)\) at the point \(x=a\) is defined as: \[f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\] Alternative description: \[f'(a) = \lim_{h\to 9}\frac{f(a+h)-f(a)}{h}\]

• Equation for tangent line:

  Tangent Line: \[y - f(a) = f'(a)(x-a)\]

• Example 1:
  • \(f(x) = x^3\), \(a=2\).
  • \(\frac{f(a+h)-f(a)}{h} = \frac{(2+h)^3-8}{h} = \frac{2^3+6h^2+12h + h^3 - 8}{h} = 6h + 12 +h^2\).
  • Taking limit as \(h\to 0\): \(f'(2) = 0 + 12 + 0 = 12\)
• Practice: Do same example using the other limit formulation.
• Example 2:
  • \(f(x) = \frac{1}{x}\), \(a=2\).
  • \(\frac{f(x)-f(a)}{x-a} = \frac{\frac{1}{x}-\frac{1}{a}}{x-a} = \frac{\frac{a-x}{xa}}{x-a} = -\frac{1}{xa} = -\frac{1}{a^2} = -\frac{1}{4}\)
• Derivatives of linear and constant equations:
  • \((mx+b)' = m\)
  • \(b' = 0\)