Finding the Derivative of a Function

Constant Functions

• The derivative of any constant is 0.
  • Example: Derivative of 5 or -7 is 0.

Concept of a Derivative

• A derivative provides the slope of a function at a given x value.
• Example: For f(x) = 8, it graphs as a straight line y=8 with a slope of 0.

Differentiation Notation

• d/dx indicates differentiation with respect to x.

Derivative of Monomials

• Power Rule: Derivative of x^n is n * x^(n-1).
  • Example: Derivative of x^2 is 2x.
  • Derivative examples:
    • x^3: 3x^2
    • x^4: 4x^3
    • x^5: 5x^4

Derivative of a Constant Times a Function

• Constant Multiple Rule: Derivative of cf(x) is cf'(x).
  • Example: Derivative of 4x^7 is 28x^6.

Examples with Constants

• Derivative of 8x^4: 32x^3
• Derivative of 5x^6: 30x^5

Limit Definition of a Derivative

• f'(x) = lim as h -> 0 of (f(x+h) - f(x))/h.
• Confirming derivative of x^2 = 2x using limits.

Derivatives of Polynomial Functions

• Differentiate each monomial separately.
  • Example: For f(x) = x^3 + 7x^2 - 8x + 6, f'(x) = 3x^2 + 14x - 8.

Slope of Tangent Lines

• Derivative provides slope of tangent at x = n.
  • Example: For f(x) = x^2, slope at x = 1 is 2.

Tangent vs. Secant Lines

• Tangent: touches curve at one point.
• Secant: touches curve at two points.

Derivative of a Rational Function

• Rewrite using negative exponents if needed.
  • Example: Derivative of 1/x is -1/x^2.

Derivative of Radical Functions

• Rewrite radicals using rational exponents.
  • Example: Derivative of √x is 1/(2√x).

Product Rule

• Formula: (f * g)' = f'g + fg'.
• Example: Derivative of x^2 * sin(x) is 2xsin(x) + x^2cos(x).

Quotient Rule

• Formula: (f/g)' = (g f' - f g') / g^2.
• Example: Derivative of (5x + 6)/(3x - 7).

Derivatives of Trigonometric Functions

• Derivative of sin(x) is cos(x).
• Derivative of cos(x) is -sin(x).
• Derivative of tan(x) is sec^2(x).
• Derivative of sec(x) is sec(x)tan(x).
• Derivative of csc(x) is -csc(x)cot(x).
• Derivative of cot(x) is -csc^2(x).

Practice Problems

• Various practice problems with derivatives of polynomials, rational functions, and trigonometric functions are discussed.
• Use of derivative rules like product and quotient rule for complex functions.

Summary

• Derivatives help find the slope of the tangent line to a curve at a specific point.
• Several rules (power, product, quotient, and trigonometric derivatives) are used to simplify the differentiation process.
• Understanding the difference between tangent and secant lines is crucial for applying derivatives correctly.