Derivatives: Product Rule, Quotient Rule, and Power Rule

Derivatives: Product Rule, Quotient Rule, and Power Rule

Introduction

This study guide covers essential rules for finding derivatives in calculus, including the Product Rule, Quotient Rule, Constant Rule, and Power Rule. These rules are fundamental for differentiating combinations of functions and are widely used in calculus problems.

The Product Rule

Definition and Formula

The Product Rule is used to find the derivative of the product of two differentiable functions.

• Formula:

$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{d}{dx}[g(x)] + g(x) \cdot \frac{d}{dx}[f(x)]$

• Key Points:

  • Both functions must be differentiable.

  • The derivative of the product is not simply the product of the derivatives.

Example:

Find the derivative of $h(x) = (x^3 - 3)(x^4 + 8)$.

• Let $f(x) = x^3 - 3$, $g(x) = x^4 + 8$

• $f'(x) = 3x^2$, $g'(x) = 4x^3$

• Apply the product rule:

$h'(x) = f'(x)g(x) + f(x)g'(x) = 3x^2(x^4 + 8) + (x^3 - 3)4x^3$

$= 3x^6 + 24x^2 + 4x^6 - 12x^3$

$= 7x^6 - 12x^3 + 24x^2$

The Quotient Rule

Definition and Formula

The Quotient Rule is used to find the derivative of the quotient of two differentiable functions.

• Formula:

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)\frac{d}{dx}[f(x)] - f(x)\frac{d}{dx}[g(x)]}{[g(x)]^2}$, where $g(x) \neq 0$

• Key Points:

  • The denominator must not be zero.

  • The order of subtraction in the numerator is important.

Example:

Find the derivative of $f(x) = \frac{x^2 - 4}{x^2 + 2}$.

• Let $f(x) = x^2 - 4$, $g(x) = x^2 + 2$

• $f'(x) = 2x$, $g'(x) = 2x$

• Apply the quotient rule:

$f'(x) = \frac{(x^2 + 2)(2x) - (x^2 - 4)(2x)}{(x^2 + 2)^2}$

$= \frac{2x(x^2 + 2 - x^2 + 4)}{(x^2 + 2)^2}$

$= \frac{2x(6)}{(x^2 + 2)^2} = \frac{12x}{(x^2 + 2)^2}$

The Constant Rule

Definition and Formula

The Constant Rule states that the derivative of a constant function is zero.

• Formula:

$\frac{d}{dx}[c] = 0$

• Key Points:

  • Applies to any real constant $c$.

  • The graph of a constant function is a horizontal line with slope zero.

The Power Rule

Definition and Formula

The Power Rule is used to differentiate functions of the form $f(x) = x^n$, where $n$ is any rational number.

• Formula:

$\frac{d}{dx}[x^n] = n x^{n-1}$

• Key Points:

  • Works for positive, negative, and fractional exponents.

  • Can be applied repeatedly for higher order derivatives.

Example:

Find the derivative of $g(u) = u^8$.

$g'(u) = 8u^{7}$

Higher Order Derivatives

Definition

Higher order derivatives are derivatives of derivatives. The $n$th derivative is denoted $f^{(n)}(x)$.

• Formula for $n$th derivative:

$f^{(n)}(x) = \frac{d^n}{dx^n}[f(x)]$

Example:

Given $f(x) = 3x^3 - 7x^{1/2}$, find $f''(x)$.

• First derivative: $f'(x) = 9x^2 - \frac{7}{2}x^{-1/2}$

• Second derivative: $f''(x) = 18x - \frac{7}{4}x^{-3/2}$

Applications: Tangent Lines and Horizontal Tangents

Finding Tangent Lines

• The equation of the tangent line to $y = f(x)$ at $x = a$ is:

$y = f(a) + f'(a)(x - a)$

Example:

Given $f(x) = x^3 - x^2 - 12x + 1$, find the equation of the tangent line at $x = 1$.

• Compute $f(1)$ and $f'(1)$, then substitute into the tangent line formula.

Horizontal Tangent Lines

• A function has a horizontal tangent where $f'(x) = 0$.

• Solve $f'(x) = 0$ for $x$ to find such points.

Summary Table: Derivative Rules

Rule	Formula	Example
Constant Rule	$\frac{d}{dx}[c] = 0$	$\frac{d}{dx}[5] = 0$
Power Rule	$\frac{d}{dx}[x^n] = n x^{n-1}$	$\frac{d}{dx}[x^4] = 4x^3$
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$	$\frac{d}{dx}[x^2 \sin x] = x^2 \cos x + 2x \sin x$
Quotient Rule	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx}\left[\frac{x}{x+1}\right] = \frac{(x+1)\cdot1 - x\cdot1}{(x+1)^2} = \frac{1}{(x+1)^2}$

Practice Problems

• Compute the derivative of $y = \frac{5}{14x^7}$.

• Find the derivative of $g(t) = 4t^{3/2}$.

• Given $f(x) = x^3 - x^2 - 12x + 1$, find all $x$ where the tangent is horizontal.

Additional info: Some steps and explanations have been expanded for clarity and completeness, including the summary table and explicit formulas for tangent lines and higher order derivatives.