Calculus I: Derivatives – Rules, Trigonometric, Exponential, and Logarithmic Functions

Objectives and Overview

This study guide reviews key derivative concepts from Calculus I, focusing on differentiation rules, trigonometric, exponential, and logarithmic functions, and the Chain Rule. Mastery of these topics is essential for solving calculus problems involving rates of change, tangent lines, and function analysis.

3.3 Differentiation Rules

Basic Differentiation Rules

• Constant Rule: The derivative of a constant is zero. $\frac{d}{dx}[c] = 0$

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

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

• Product Rule: The derivative of a product of two functions: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: The derivative of a quotient of two functions: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Higher Order Derivatives: The second derivative is the derivative of the first derivative, denoted $f''(x)$.

• Tangent Line Slope: The derivative at a point gives the slope of the tangent line to the curve at that point.

3.5 Derivatives of Trigonometric Functions

Trigonometric Derivatives

• Basic Trig Derivatives:

  • $\frac{d}{dx}[\sin x] = \cos x$

  • $\frac{d}{dx}[\cos x] = -\sin x$

  • $\frac{d}{dx}[\tan x] = \sec^2 x$

  • $\frac{d}{dx}[\cot x] = -\csc^2 x$

  • $\frac{d}{dx}[\sec x] = \sec x \tan x$

  • $\frac{d}{dx}[\csc x] = -\csc x \cot x$

• Applications: Use these derivatives to find slopes, graph tangent lines, and solve conceptual questions involving trigonometric functions.

• Example: Find the derivative of $f(x) = \sin x + \cos x$. $f'(x) = \cos x - \sin x$

3.6 The Chain Rule

Chain Rule for Composite Functions

• Definition: The Chain Rule is used to differentiate composite functions of the form $h(x) = f(g(x))$.

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

• Example: If $y = \sin(x^2)$, then $\frac{dy}{dx} = \cos(x^2) \cdot 2x$.

3.3 Derivatives of Exponential Functions and 3.8 Derivatives of Logarithmic Functions

Exponential Functions

• Natural Exponential Function: $\frac{d}{dx}[e^x] = e^x$

• General Exponential Function: $\frac{d}{dx}[a^x] = a^x \ln a$

• Example: $\frac{d}{dx}[2^x] = 2^x \ln 2$

Logarithmic Functions

• Natural Logarithm: $\frac{d}{dx}[\ln x] = \frac{1}{x}$

• Logarithm with Base $a$: $\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$

• Example: $\frac{d}{dx}[\log_{10} x] = \frac{1}{x \ln 10}$

Properties and Conversions

• Logarithm Properties:

  • $\log_a (xy) = \log_a x + \log_a y$

  • $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

  • $\log_a (x^r) = r \log_a x$

  • $\log_a x = \frac{\ln x}{\ln a}$

• Exponential-Logarithmic Conversion: $a^x = e^{x \ln a}$ and $\log_a x = \frac{\ln x}{\ln a}$

Summary Table: Common Derivatives

Function	Derivative
$c$ (constant)	$0$
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\frac{1}{x}$
$\log_a x$	$\frac{1}{x \ln a}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

Assignments and Practice

• Review the above rules and derivatives before attempting homework assignments.

• Practice finding derivatives using the rules for sums, products, quotients, and compositions (Chain Rule).

• Apply these concepts to graphing tangent lines and solving application problems.