Differentiation 2

Study Notes

Differentiation 2 covers advanced techniques for finding derivatives, including the chain rule, derivatives of exponential and logarithmic functions, and optimization.

• The Chain Rule — a method for differentiating composite functions. Example: If $y = (3x^2 + 2)^5$, use the chain rule to find $\frac{dy}{dx}$.
• Derivatives of Exponential Functions — finding the derivative of functions with bases that are constants raised to a variable power. Example: The derivative of $e^x$ is $e^x$.
• Derivatives of Logarithmic Functions — finding the derivative of logarithmic functions. Example: The derivative of $\ln(x)$ is $\frac{1}{x}$.
• The Product Rule — a method for differentiating products of two functions. Example: If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$.
• The Quotient Rule — a method for differentiating quotients of two functions. Example: If $y = \frac{u(x)}{v(x)}$, then $\frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$.
• Optimization — using derivatives to find maximum or minimum values of functions. Example: Find the maximum area of a rectangle with a fixed perimeter.

Exam Tips

Key Definitions to Remember

• The Chain Rule
• Derivatives of Exponential Functions
• Derivatives of Logarithmic Functions
• The Product Rule
• The Quotient Rule
• Optimization

Common Confusions

• Mixing up the product and quotient rules
• Forgetting to apply the chain rule in composite functions
• Incorrectly simplifying derivatives of logarithmic functions

Typical Exam Questions

• How do you apply the chain rule to $y = (2x + 3)^4$? Differentiate the outer function and multiply by the derivative of the inner function.
• What is the derivative of $e^{3x}$? $3e^{3x}$
• How do you find the maximum value of a function using derivatives? Set the derivative equal to zero and solve for critical points.

What Examiners Usually Test

• Ability to correctly apply differentiation rules
• Understanding of when to use each rule
• Problem-solving skills in optimization scenarios