Differentiation Rules and Applications in Calculus

Differentiation Rules in Calculus

Introduction

Differentiation is a fundamental concept in calculus, used to determine the rate at which a function changes. The following notes summarize the main rules for differentiating functions, including the power, product, and quotient rules, as well as derivatives of exponential and trigonometric functions. Examples are provided to illustrate each rule.

Derivative of a Constant Function

• Definition: The derivative of a constant function is always zero.

• Formula: $\frac{d}{dx}(c) = 0$

• Explanation: Since a constant does not change, its rate of change is zero.

The Power Rule

• Definition: Used to differentiate functions of the form $x^n$ where $n$ is a real number.

• Formula: $\frac{d}{dx}(x^n) = n x^{n-1}$

• Example: $\frac{d}{dx}(x^6) = 6x^5$

• General Version: Applies for any real exponent, including fractional and negative powers. $\frac{d}{dx}(x^n) = n x^{n-1}$

• Example: $\frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2}$

The Constant Multiple Rule

• Definition: The derivative of a constant times a function is the constant times the derivative of the function.

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

• Example: $\frac{d}{dx}(2x^3) = 2 \cdot 3x^2 = 6x^2$

The Sum and Difference Rules

• Definition: The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives.

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

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

• Example: $\frac{d}{dx}(x^2 + e^x) = 2x + e^x$

Derivative of the Natural Exponential Function

• Definition: The derivative of $e^x$ is $e^x$.

• Formula: $\frac{d}{dx}(e^x) = e^x$

• Example: $\frac{d}{dx}(e^{x^2}) = 2x e^{x^2}$ (using the chain rule; additional info)

Finding the Derivative: Worked Examples

• Example 1: $r(t) = \frac{a}{t^2} + \frac{b}{t^4}$ $r'(t) = -2a t^{-3} - 4b t^{-5}$

• Example 2: $h(w) = \sqrt{w} - \sqrt{2}$ $h'(w) = \frac{1}{2} w^{-1/2}$

• Example 3: $y(t) = 4e^{t^2} - \frac{2}{\sqrt{t}}$ $y'(t) = 8t e^{t^2} + t^{-3/2}$

Equation of the Tangent Line

• Definition: The tangent line to a curve at a point $(a, f(a))$ has slope $f'(a)$.

• Formula: $y - f(a) = f'(a)(x - a)$

• Example: For $y = x + \frac{2}{x}$ at $(2, 3)$: $y' = 1 - 2x^{-2}$ $y'(2) = 1 - 2(2^{-2}) = 1 - 2(1/4) = 1 - 0.5 = 0.5$ Equation: $y - 3 = \frac{1}{2}(x - 2)$

The Product Rule

• Definition: Used to differentiate the product of two functions.

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

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

The Quotient Rule

• Definition: Used to differentiate the quotient of two functions.

• Formula: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$

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

Derivatives of Trigonometric Functions

The following table summarizes the derivatives of the six basic trigonometric functions:

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$(\sec x)^2$
$\csc x$	$-\csc x \cot x$
$\sec x$	$\sec x \tan x$
$\cot x$	$-\csc^2 x$

• Example: Find the derivative of $y = \frac{\cos x}{1 + 2 \sin x}$ Using the quotient rule: $y' = \frac{(1 + 2 \sin x)(-\sin x) - (\cos x)(2 \cos x)}{(1 + 2 \sin x)^2}$

Summary Table: Differentiation Rules

Rule	Formula	Example
Constant	$\frac{d}{dx}(c) = 0$	$\frac{d}{dx}(5) = 0$
Power	$\frac{d}{dx}(x^n) = n x^{n-1}$	$\frac{d}{dx}(x^3) = 3x^2$
Constant Multiple	$\frac{d}{dx}[c f(x)] = c f'(x)$	$\frac{d}{dx}(2x^2) = 4x$
Sum/Difference	$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$	$\frac{d}{dx}(x^2 + e^x) = 2x + e^x$
Product	$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$	$\frac{d}{dx}(x^2 e^x) = x^2 e^x + 2x e^x$
Quotient	$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx}\left(\frac{x^2}{x+1}\right)$
Exponential	$\frac{d}{dx}(e^x) = e^x$	$\frac{d}{dx}(e^{2x}) = 2e^{2x}$
Trigonometric	See table above	$\frac{d}{dx}(\sin x) = \cos x$

Conclusion

Mastering these differentiation rules is essential for solving calculus problems involving rates of change, tangent lines, and the behavior of functions. Practice applying these rules to a variety of functions to build proficiency.