Differentiation Rules and Applications in Calculus

Differentiation Rules

Basic Differentiation Rules

The process of differentiation allows us to find the rate of change of a function with respect to its variable. Several fundamental rules simplify the calculation of derivatives for common types of functions.

• Derivative of a Constant Function: The derivative of any constant function is zero. $\frac{d}{dx}c = 0$ where $c$ is a constant.

• Power Rule: For any real number $n$, the derivative of $x^n$ is: $\frac{d}{dx}x^n = n x^{n-1}$

• Constant Multiple Rule: If $c$ is a constant and $f(x)$ is differentiable: $\frac{d}{dx}[c f(x)] = c \frac{d}{dx}f(x)$

• Sum and Difference Rules: The derivative of a sum or difference is the sum or difference of the derivatives: $\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$

Examples of Basic Differentiation

• Example 1: $f(x) = 205$ $f'(x) = 0$

• Example 2: $g(t) = 5t + 4t^2$ $g'(t) = 5 + 8t$

• Example 3: $f(x) = x^{3/2} + x^{-3}$ $f'(x) = \frac{3}{2}x^{1/2} - 3x^{-4}$

Derivative of Exponential Functions

Exponential Differentiation

Exponential functions have unique differentiation properties, especially the natural exponential function $e^x$.

• Derivative of $e^x$: $\frac{d}{dx}e^x = e^x$

• Derivative of $c \cdot e^x$: $\frac{d}{dx}[c \cdot e^x] = c \cdot e^x$

Example

• Example: $f(x) = e^x + 4x^2 + 3$ $f'(x) = e^x + 8x$

Equations of Tangent Lines

Finding Tangent Lines

The tangent line to a curve at a given point is found using the derivative to determine the slope at that point.

• General Formula: For a function $y = f(x)$ at point $(a, f(a))$, the tangent line is: $y - f(a) = f'(a)(x - a)$

• Example: $y = 2x^3 - x^2 + 2$ at $(1, 3)$ $y' = 6x^2 - 2x$ $y'(1) = 6(1)^2 - 2(1) = 4$ Tangent line: $y - 3 = 4(x - 1)$

Horizontal Tangents

Finding Horizontal Tangents

A tangent line is horizontal where the derivative equals zero.

• Example: $y = x^3 - 3x - 2$ $y' = 3x^2 - 3$ Set $y' = 0$: $3x^2 - 3 = 0 \Rightarrow x = \pm 1$

Product Rule

Product Rule for Differentiation

The product rule is used when differentiating the product of two functions.

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$

• Example: $y = (4x^2 + 3)(2x + 5)$ $y' = (4x^2 + 3) \cdot 2 + (2x + 5) \cdot 8x = 8x^2 + 6 + 16x^2 + 40x$

Quotient Rule

Quotient Rule for Differentiation

The quotient rule is used when differentiating the quotient of two functions.

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$

• Example: $G(x) = \frac{6x^4 - 5}{x + 1}$ $G'(x) = \frac{(x + 1) \cdot 24x^3 - (6x^4 - 5) \cdot 1}{(x + 1)^2}$

Higher Derivatives

Second and Higher Derivatives

The derivative of a derivative is called a higher derivative. The second derivative measures the rate of change of the rate of change.

• Notation: $f''(x) = \frac{d^2y}{dx^2}$

• Example: $f(x) = 6e^x - 2x^2 + 3$ $f'(x) = 6e^x - 4x$ $f''(x) = 6e^x - 4$

Summary Table: Differentiation Rules

Rule	Formula	Example
Constant	$\frac{d}{dx}c = 0$	$f(x) = 5 \Rightarrow f'(x) = 0$
Power	$\frac{d}{dx}x^n = n x^{n-1}$	$f(x) = x^3 \Rightarrow f'(x) = 3x^2$
Sum/Difference	$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$	$f(x) = x^2 + x \Rightarrow f'(x) = 2x + 1$
Product	$\frac{d}{dx}[u v] = u v' + v u'$	$f(x) = x^2 \cdot x \Rightarrow f'(x) = x^2 \cdot 1 + x \cdot 2x = x^2 + 2x^2$
Quotient	$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v u' - u v'}{v^2}$	$f(x) = \frac{x^2}{x} \Rightarrow f'(x) = \frac{x \cdot 2x - x^2 \cdot 1}{x^2}$
Exponential	$\frac{d}{dx}e^x = e^x$	$f(x) = e^x \Rightarrow f'(x) = e^x$