Derivative Rules

Reading

• Sections 3.3

Practice problems

• Section 3.3: 3, 7, 11, 15, 17, 21, 27, 31, 37
• To turn in (together with 3.5): 3.3 4, 18, 32

Notes

Derivative Rules

• Product Rule:

  Product Rule \[(fg)'(x) = f'(x)g(x) + f(x) g'(x)\]

• Examples:
  • \((x\sin x)' = \sin x + x\cos x\)
  • \((x^2)' = (x\cdot x)' = x'\cdot x + x \cdot x' = 1\cdot x + x\cdot 1 = 2x\)
  • \(x^2(\cos x+\sin x)\)
• Proof. Visualizing the difference of products in the limit

• Quotient Rule:

  Quotient Rule \[\left(\frac{f}{g}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\]

• Examples:
  • \(\frac{1}{x}\)
  • \(\frac{x}{1+x+x^2}\)