The Product Rule and Quotient Rule

Introduction

Key Concepts

The Product Rule

The Product Rule is a fundamental theorem in differential calculus used to find the derivative of the product of two functions. If you have two differentiable functions, say $u(x)$ and $v(x)$, then the derivative of their product is given by:

$$ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) $$

This rule states that to differentiate the product of two functions, you differentiate the first function and multiply it by the second function, then add the product of the first function and the derivative of the second function.

Example of the Product Rule

Consider the functions $u(x) = x^2$ and $v(x) = \sin(x)$. Using the Product Rule:

$$ \frac{d}{dx}[x^2 \cdot \sin(x)] = 2x \cdot \sin(x) + x^2 \cdot \cos(x) $$

Thus, the derivative of $x^2 \cdot \sin(x)$ is $2x \cdot \sin(x) + x^2 \cdot \cos(x)$.

The Quotient Rule

The Quotient Rule is another essential derivative rule used when differentiating a function that is the quotient of two differentiable functions. If $f(x) = \frac{u(x)}{v(x)}$, then the derivative of $f(x)$ is:

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

This formula indicates that the derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the square of the denominator.

Example of the Quotient Rule

Let’s differentiate $f(x) = \frac{e^x}{x^3}$. Here, $u(x) = e^x$ and $v(x) = x^3$. Applying the Quotient Rule:

$$ f'(x) = \frac{e^x \cdot x^3 - e^x \cdot 3x^2}{x^6} = \frac{e^x (x^3 - 3x^2)}{x^6} = e^x \cdot \frac{x - 3}{x^4} $$>

Therefore, the derivative of $\frac{e^x}{x^3}$ is $e^x \cdot \frac{x - 3}{x^4}$.

Applications of the Product and Quotient Rules

• Physics: Calculating rates of change in motion, such as velocity and acceleration when position is given by a product or quotient of functions.
• Economics: Determining marginal costs and revenues where cost functions are products or ratios of multiple economic factors.
• Engineering: Analyzing systems where output depends on the product or ratio of various system parameters.

Common Mistakes and How to Avoid Them

• Sign Errors: Ensure that you carefully handle the subtraction in the Quotient Rule to avoid sign mistakes.
• Incorrect Differentiation: Double-check the derivatives of the individual functions before applying the Product or Quotient Rule.
• Algebraic Simplification: After applying the rules, simplify the expression correctly to avoid computational errors.

Higher-Order Applications

While the Product and Quotient Rules are primarily used for first derivatives, they also extend to higher-order derivatives through repeated application. For instance, finding the second derivative of a product of functions requires applying the Product Rule to the first derivative expression.

Visual Understanding

Graphically, the Product Rule helps in understanding how the slope of a product of two functions behaves based on the slopes and values of the individual functions. Similarly, the Quotient Rule provides insight into how the rate of change of a ratio of functions is influenced by both the numerator and the denominator.

Proof of the Product Rule

The Product Rule can be derived using the definition of the derivative:

$$ \frac{d}{dx}[u(x) \cdot v(x)] = \lim_{h \to 0} \frac{u(x+h)v(x+h) - u(x)v(x)}{h} $$>

Expanding and re-arranging terms:

$$ = \lim_{h \to 0} \left[ u(x) \frac{v(x+h) - v(x)}{h} + v(x+h) \frac{u(x+h) - u(x)}{h} \right] $$>

Taking the limit as $h$ approaches 0:

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

This completes the proof of the Product Rule.

Proof of the Quotient Rule

The Quotient Rule can similarly be derived from the definition of the derivative:

$$ f'(x) = \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right ) = \lim_{h \to 0} \frac{\frac{u(x+h)}{v(x+h)} - \frac{u(x)}{v(x)}}{h} $$>

Combine the fractions:

$$ = \lim_{h \to 0} \frac{u(x+h)v(x) - u(x)v(x+h)}{h \cdot v(x+h) \cdot v(x)} $$>

Add and subtract $u(x)v(x)$ in the numerator:

$$ = \lim_{h \to 0} \frac{u(x+h)v(x) - u(x)v(x) + u(x)v(x) - u(x)v(x+h)}{h \cdot v(x+h) \cdot v(x)} $$>

Factor terms:

$$ = \lim_{h \to 0} \left[ \frac{u(x+h) - u(x)}{h} \cdot v(x) - u(x) \cdot \frac{v(x+h) - v(x)}{h} \right ] \cdot \frac{1}{v(x) \cdot v(x+h)} $$>

Taking the limit as $h$ approaches 0:

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

This concludes the proof of the Quotient Rule.

Derivatives Using the Product and Quotient Rules

• Example 1: Find the derivative of $f(x) = (3x^2 + 2x)(\ln x)$.
• Solution: Let $u(x) = 3x^2 + 2x$ and $v(x) = \ln x$. Then:

$$ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) = (6x + 2) \cdot \ln x + (3x^2 + 2x) \cdot \frac{1}{x} $$ $$ = (6x + 2) \ln x + 3x + 2 $$

• Example 2: Differentiate $g(x) = \frac{e^x}{x^2 + 1}$.
• Solution: Let $u(x) = e^x$ and $v(x) = x^2 + 1$. Then:

$$ g'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2} = \frac{e^x (x^2 + 1) - e^x (2x)}{(x^2 + 1)^2} $$ $$ = \frac{e^x (x^2 + 1 - 2x)}{(x^2 + 1)^2} = e^x \cdot \frac{x^2 - 2x + 1}{(x^2 + 1)^2} $$

Extended Applications

Beyond basic differentiation, the Product and Quotient Rules are instrumental in solving problems involving implicit differentiation and higher-order derivatives. They also play a critical role in optimizing functions in various fields such as physics, engineering, and economics.

Tips for Mastering the Product and Quotient Rules

• Practice: Regularly solve a variety of problems to become comfortable with applying these rules in different contexts.
• Understand the Theory: Grasp the underlying principles behind these rules to apply them effectively, especially in complex scenarios.
• Memorize the Formulas: While understanding is crucial, having the formulas at your fingertips enables quicker problem-solving.
• Check Your Work: Always verify each step to minimize errors, particularly in algebraic manipulation after applying the rules.

Integration with Other Derivative Rules

The Product and Quotient Rules often work in tandem with other derivative rules such as the Chain Rule and the Power Rule. For instance, when differentiating composite functions, the Chain Rule may be employed in conjunction with the Product or Quotient Rules to simplify the process.

Comparison Table

Aspect	Product Rule	Quotient Rule
Definition	Derivative of a product of two functions.	Derivative of a quotient of two functions.
Formula	$u'(x)v(x) + u(x)v'(x)$	$\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$
Application	Used when functions are multiplied.	Used when functions are divided.
Complexity	Simpler as it involves addition.	More complex due to subtraction and division.
Common Mistakes	Forgetting to apply the rule to both functions.	Incorrect sign in the numerator or forgetting to square the denominator.
Example	$\frac{d}{dx}[x^2 \cdot \sin x] = 2x \sin x + x^2 \cos x$	$\frac{d}{dx}\left[\frac{e^x}{x^3}\right] = \frac{e^x(x^3 - 3x^2)}{x^6}$

Summary and Key Takeaways