Lecture on Derivatives

Introduction to Derivatives

• Definition: A derivative is a function that gives the slope at any x value.
• Derivative of Constants: Always 0. Example: Derivative of 5 is 0.

Derivatives of Monomials

• Power Rule: Derivative of ( x^n ) is ( n \times x^{n-1} ).
  • Example for ( x^2 ): ( 2x )
  • Example for ( x^3 ): ( 3x^2 )
  • Example for ( x^5 ): ( 5x^4 )

Constant Multiple Rule

• Derivative of a constant multiplied by a function is the constant times the derivative of the function.
  • Example: ( 4x^7 ) becomes ( 28x^6 ) after applying the rule.

Finding the Derivative Using Limits

• Definition of Derivative: ( f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} )
  • Used to prove derivatives of functions like ( x^2 ).

Tangent and Secant Lines

• Tangent Line: Touches the curve at a single point.
• Secant Line: Touches the curve at two points.
• Slope of Tangent Line: Given by the derivative at a specific x value.

Derivatives of Polynomial Functions

• Differentiate each term individually using the power rule.
• Example: ( f(x) = x^3 + 7x^2 - 8x + 6 ) results in ( f'(x) = 3x^2 + 14x - 8 )

Derivatives of Rational and Radical Functions

• Rewrite using negative exponents or rational exponents, then apply power rule.
  • Example: ( 1/x ) becomes ( -1/x^2 )
  • Example: ( \sqrt{x} ) becomes ( 1/(2\sqrt{x}) )

Chain Rule (Foreshadowed)

• Method for handling derivatives of composite functions.
• Example Exercise: Expand expressions like ((2x - 3)^2) and then differentiate.

Product Rule

• Formula: ((fg)' = f'g + fg')
• Example: Derivative of ( x^2 \sin x ) results in ( 2x \sin x + x^2 \cos x )

Quotient Rule

• Formula: ((\frac{f}{g})' = \frac{g f' - f g'}{g^2})
• Example: Derivative of ( \frac{5x+6}{3x-7} )

Derivatives of Trigonometric Functions

• ( \frac{d}{dx} \sin x = \cos x )
• ( \frac{d}{dx} \cos x = -\sin x )
• Other derivatives of trig functions like secant, cosecant, tangent, and cotangent.

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Key Takeaways

• Derivative gives the slope of tangent lines.
• Power, product, and quotient rules are essential for finding derivatives.
• Understanding trigonometric derivatives is crucial for calculus problems.
• The definition of derivative via limits is foundational in calculus.

This concludes the lecture notes on derivatives.