Differentiation Techniques and Applications in Calculus

§1.5 - Differentiation Techniques: The Power and Sum-Difference Rules

Leibniz Notation

Leibniz notation is a standard way to denote derivatives in calculus. If y = f(x), the derivative of y with respect to x is written as $\frac{dy}{dx} = y' = f'(x)$. The derivative evaluated at a specific value can be written as $f'(2)$ or $\left.\frac{dy}{dx}\right|_{x=2}$.

• Derivative Operator: $\frac{d}{dx}$ placed next to an expression denotes its derivative.

• Examples:

  • $\frac{d}{dx}x^2 = 2x$

  • $\frac{d}{dx}x^3 = 3x^2$

  • $\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}$

The Power Rule

The Power Rule is a fundamental technique for differentiating functions of the form $x^n$.

• Theorem: For any real number n, if $y = x^n$, then $\frac{d}{dx}x^n = n \cdot x^{n-1}$.

• Examples:

  • $\frac{d}{dx}x^5 = 5x^4$

  • $\frac{d}{dx}x = 1$

  • $\frac{d}{dx}x^{-4} = -4x^{-5}$

  • $\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}$

The Derivative of a Constant Function

The derivative of a constant function is always zero, as the slope of its graph is flat.

• Theorem: $\frac{d}{dx}c = 0$ for any constant c.

The Derivative of a Constant Times a Function

When differentiating a constant times a function, the constant can be factored out.

• Theorem: $\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}f(x)$

• Examples:

  • $\frac{d}{dx}7x^4 = 28x^3$

  • $\frac{d}{dx}(-9x) = -9$

  • $\frac{d}{dx}\left(\frac{1}{5}x^2\right) = \frac{2}{5}x$

The Derivative of a Sum or a Difference

The derivative of a sum or difference of functions is the sum or difference of their derivatives.

• Theorem:

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

  • Difference: $\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}f(x) - \frac{d}{dx}g(x)$

• Example: $\frac{d}{dx}(5x^3 - 7) = 15x^2$

Slopes of Tangent Lines

To find points where the tangent line has a specific slope, set the derivative equal to the desired value and solve for x.

• Example: For $f(x) = -x^3 + 6x^2$, find where the tangent is horizontal (set $f'(x) = 0$) or has slope 9 (set $f'(x) = 9$).

Applications: Population Growth and Ticket Prices

Derivatives are used to model rates of change in real-world scenarios, such as population growth and pricing.

• Example: Population growth rate: $P(t) = 100,000 + 2000t^2$, $\frac{dP}{dt} = 4000t$.

• Example: Super Bowl ticket prices: $p(t) = 0.858t^2 - 18.864t + 78.354$.

§1.6 - Differentiation Techniques: The Product and Quotient Rules

The Product Rule

The Product Rule is used to differentiate products of two functions.

• Theorem: If $F(x) = f(x) \cdot g(x)$, then $F'(x) = f(x) \cdot g'(x) + g(x) \cdot f'(x)$.

• Example: $\frac{d}{dx}[(x^4 - 2x^3 - 7)(3x^2 - 5x)]$

The Quotient Rule

The Quotient Rule is used to differentiate quotients of two functions.

• Theorem: If $Q(x) = \frac{f(x)}{g(x)}$, then $Q'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$

• Example: $\frac{d}{dx}\left(\frac{x^5}{x^2}\right)$

Application: Average Cost, Revenue, and Profit

In business, average cost, revenue, and profit are important for analyzing performance. These are defined as follows:

Quantity	Formula
Average Cost	$\frac{C(x)}{x}$
Average Revenue	$\frac{R(x)}{x}$
Average Profit	$\frac{P(x)}{x}$

§1.7 - Differentiation Techniques: The Chain Rule

The Extended Power Rule

The Extended Power Rule generalizes the Power Rule to functions raised to a power.

• Theorem: If $y = [g(x)]^n$, then $\frac{d}{dx}[g(x)]^n = n[g(x)]^{n-1} \cdot g'(x)$

• Example: $\frac{d}{dx}(1 + x^3)^5 = 5(1 + x^3)^4 \cdot 3x^2$

Composition of Functions and the Chain Rule

The Chain Rule is used to differentiate compositions of functions.

• Definition: The composition $f \circ g$ is $f(g(x))$.

• Theorem: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

• Example: If $f(x) = x^4$ and $g(x) = 2 + x^3$, then $\frac{d}{dx}[f(g(x))] = 4(g(x))^3 \cdot g'(x)$

§1.8 - Higher Order Derivatives

Higher Order Derivatives

Higher order derivatives are derivatives of derivatives, representing rates of change of rates of change.

• Notation: $f''(x)$ is the second derivative, $f'''(x)$ is the third, and $f^{(n)}(x)$ is the nth derivative.

• Leibniz Notation: $\frac{d^2y}{dx^2}$, $\frac{d^3y}{dx^3}$, etc.

• Example: For $y = x^5 - 3x^4 + x$, $\frac{d^2y}{dx^2} = 20x^3 - 36x^2$

Velocity and Acceleration

When a function describes position over time, its first derivative is velocity and its second derivative is acceleration.

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• Example: For $s(t) = 4.905t^2$, $v(t) = 9.81t$, $a(t) = 9.81$

§2.2 - Derivatives of Exponential (Base-e) Functions

Derivatives of Exponential Functions

The exponential function with base e, $f(x) = e^x$, is unique in that its derivative is itself.

• Theorem: $\frac{d}{dx}e^x = e^x$

• Chain Rule for Exponential Functions: $\frac{d}{dx}e^{f(x)} = e^{f(x)} \cdot f'(x)$

• Example: $\frac{d}{dx}e^{8x} = 8e^{8x}$

§2.3 - Derivatives of Natural Logarithmic Functions

Derivatives of Natural Logarithmic Functions

The natural logarithm function, $\ln x$, has a simple derivative.

• Theorem: $\frac{d}{dx}\ln x = \frac{1}{x}$

• Chain Rule for Logarithms: $\frac{d}{dx}\ln f(x) = \frac{1}{f(x)} \cdot f'(x)$

• Example: $\frac{d}{dx}\ln(x^2 - 5) = \frac{2x}{x^2 - 5}$

Applications

Logarithmic functions are used in models of learning, advertising, and other social sciences.

• Example: Learning model: $R(t) = 80 - 27\ln t$

• Example: Advertising response: $N(a) = 1000 + 200\ln a$