Rules of Differentiation (Power, Product, Quotient, Chain Rule)

Introduction

Key Concepts

1. Power Rule

Definition: If $f(x) = x^n$, then its derivative is given by:

$$f'(x) = nx^{n-1}$$

Application: The power rule simplifies the process of finding derivatives without resorting to first principles. It's applicable to polynomial functions and serves as a building block for more complex differentiation techniques.

Example:

Differentiate $f(x) = 5x^4$.

Solution:

$$f'(x) = 5 \cdot 4x^{4-1} = 20x^3$$

Key Points:

• The power rule applies to any real exponent.
• It simplifies differentiation of monomial functions.
• Essential for higher-order differentiation.

2. Product Rule

Definition: If $f(x) = u(x) \cdot v(x)$, then the derivative is:

$$f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$$

Application: The product rule is particularly useful when dealing with functions that are multiplied together, such as polynomial terms or complex expressions where expanding is cumbersome.

Example:

Differentiate $f(x) = (3x^2)(\sin x)$.

Solution:

$$f'(x) = (6x) \cdot \sin x + 3x^2 \cdot \cos x = 6x \sin x + 3x^2 \cos x$$

Key Points:

• Necessary for functions that are products of differentiable functions.
• Does not require expansion of the product.
• Can be extended to products of more than two functions.

3. Quotient Rule

Definition: If $f(x) = \frac{u(x)}{v(x)}$, then the derivative is:

$$f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2}$$

Application: The quotient rule is essential when dealing with rational functions or any scenario where one function divides another, ensuring accurate differentiation even when the denominator is a complex expression.

Example:

Differentiate $f(x) = \frac{e^x}{x^2}$.

Solution:

$$u(x) = e^x \Rightarrow u'(x) = e^x$$ $$v(x) = x^2 \Rightarrow v'(x) = 2x$$ $$f'(x) = \frac{e^x \cdot x^2 - e^x \cdot 2x}{x^4} = \frac{e^x (x^2 - 2x)}{x^4} = e^x \cdot \frac{x^2 - 2x}{x^4} = e^x \cdot \frac{1 - 2/x}{x^2}$$

Key Points:

• Applicable to any differentiable numerator and denominator.
• Ensures the denominator is not zero to avoid undefined expressions.
• Can be simplified further after applying the rule.

4. Chain Rule

Definition: If $f(x) = g(h(x))$, then the derivative is:

$$f'(x) = g'(h(x)) \cdot h'(x)$$

Application: The chain rule is widely used in scenarios where functions are composed of multiple layers, such as trigonometric functions of exponential functions or logarithms of polynomial functions.

Example:

Differentiate $f(x) = \sin(3x^2 + 2x)$.

Solution:

$$g(u) = \sin u \Rightarrow g'(u) = \cos u$$ $$h(x) = 3x^2 + 2x \Rightarrow h'(x) = 6x + 2$$ $$f'(x) = \cos(3x^2 + 2x) \cdot (6x + 2) = (6x + 2) \cos(3x^2 + 2x)$$

Key Points:

• Essential for differentiating nested or composite functions.
• Requires identifying the inner and outer functions.
• Can be extended to multiple layers of composition.

Comparison Table

Rule	Formula	When to Use
Power Rule	$f(x) = x^n \Rightarrow f'(x) = nx^{n-1}$	When differentiating monomial functions.
Product Rule	$f(x) = u(x)v(x) \Rightarrow f'(x) = u'(x)v(x) + u(x)v'(x)$	When differentiating the product of two functions.
Quotient Rule	$f(x) = \frac{u(x)}{v(x)} \Rightarrow f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$	When differentiating the ratio of two functions.
Chain Rule	$f(x) = g(h(x)) \Rightarrow f'(x) = g'(h(x)) \cdot h'(x)$	When differentiating composite functions.

Summary and Key Takeaways