Derivatives of Quotient Functions

Quotient Rule for Derivatives

The quotient rule is used to find the derivative of a function that is the ratio of two differentiable functions. If , then the derivative is given by:

• Formula:

• Key Points:

• Differentiate the numerator and denominator separately.

• Subtract the product of the numerator and the derivative of the denominator from the product of the derivative of the numerator and the denominator.

• Divide the result by the square of the denominator.

Example: Find the derivative of .

Simplified:

Quotient Rule with Trigonometric Functions

When applying the quotient rule to trigonometric functions, use the derivatives of sine and cosine:

Example: Find the derivative of .

Simplified (as highlighted in the notes):

Derivatives of Logarithmic and Power Functions

Derivative of Logarithmic Functions

The derivative of the natural logarithm function is:

For more complex logarithmic expressions, use the chain rule and properties of logarithms.

Derivative of Power Functions

The derivative of a power function is:

Worked Example: Derivative of a Sum Involving Powers and Logarithms

Given , find .

• Differentiate each term separately:

• (since is a constant)

Final Answer:

Logarithmic Differentiation

Using Logarithmic Properties for Differentiation

Logarithmic differentiation is useful for functions involving products, quotients, or powers. The properties of logarithms help simplify differentiation:

Example: For , the derivative is:

Summary Table: Common Derivative Rules

Additional info: Some steps and simplifications were inferred from context and standard calculus rules, as the original notes contained shorthand and partial calculations.