Calculus Study Guide: Derivatives, Rules, and Applications

Power Rule and Its Variations

Standard Power Rule

The power rule is a fundamental technique for finding derivatives when the base is the variable x.

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

• Application: Used for differentiating polynomials and any function where the variable is raised to a constant power.

• Example: $\frac{d}{dx}x^3 = 3x^2$

Power Rule with Variable Base and Chain Rule

When the base is a function of x (not just x itself), the chain rule must be applied.

• Formula: $\frac{d}{dx}[u(x)]^n = n[u(x)]^{n-1} \cdot u'(x)$

• Application: Used for composite functions where the inner function is not simply x.

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

Derivatives of Trigonometric Functions

Basic Trigonometric Derivatives

There are six primary trigonometric functions, each with a standard derivative.

• $\frac{d}{dx}\sin(x) = \cos(x)$

• $\frac{d}{dx}\cos(x) = -\sin(x)$

• $\frac{d}{dx}\tan(x) = \sec^2(x)$

• $\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)$

• $\frac{d}{dx}\sec(x) = \sec(x)\tan(x)$

• $\frac{d}{dx}\cot(x) = -\csc^2(x)$

Trigonometric Derivatives with Chain Rule

When the argument of the trigonometric function is itself a function of x, use the chain rule.

• $\frac{d}{dx}\sin(u(x)) = \cos(u(x)) \cdot u'(x)$

• $\frac{d}{dx}\cos(u(x)) = -\sin(u(x)) \cdot u'(x)$

• $\frac{d}{dx}\tan(u(x)) = \sec^2(u(x)) \cdot u'(x)$

• $\frac{d}{dx}\csc(u(x)) = -\csc(u(x))\cot(u(x)) \cdot u'(x)$

• $\frac{d}{dx}\sec(u(x)) = \sec(u(x))\tan(u(x)) \cdot u'(x)$

• $\frac{d}{dx}\cot(u(x)) = -\csc^2(u(x)) \cdot u'(x)$

Product and Quotient Rules

Product Rule

The product rule is used to differentiate the product of two functions.

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

• Example: $\frac{d}{dx}[x^2 \sin(x)] = 2x\sin(x) + x^2\cos(x)$

Quotient Rule

The quotient rule is used to differentiate the ratio of two functions.

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

• Example: $\frac{d}{dx}\left[\frac{x^2}{\sin(x)}\right] = \frac{2x\sin(x) - x^2\cos(x)}{\sin^2(x)}$

L'Hôpital's Rule

Indeterminate Forms and Limits

L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

• Formula: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the original limit is indeterminate)

• Application: Differentiate numerator and denominator separately, then take the limit.

• Example: $\lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{\cos(x)}{1} = 1$

Physics Applications: Motion and Acceleration

Distance, Velocity, and Acceleration

Calculus is used to relate position, velocity, and acceleration in physics.

• Distance (s): The position of an object as a function of time.

• Velocity (v): The derivative of position with respect to time: $v = \frac{ds}{dt}$

• Acceleration (a): The derivative of velocity with respect to time: $a = \frac{dv}{dt}$

• Example: If $s(t) = t^2$, then $v(t) = 2t$, $a(t) = 2$

Projectile Motion

For objects in free fall or projectile motion, the following equations are commonly used:

• $h = h_0 + v_0 t - \frac{1}{2}gt^2$

• $v = v_0 - gt$

• $a = -g$

• Where: $h_0$ is initial height, $v_0$ is initial velocity, $g$ is acceleration due to gravity.

Pythagorean Identities and Reciprocal Identities

Pythagorean Identities

These identities relate the squares of sine, cosine, and other trigonometric functions.

Identity	Equation
Basic	$\sin^2(x) + \cos^2(x) = 1$
Secant-Tangent	$1 + \tan^2(x) = \sec^2(x)$
Cosecant-Cotangent	$1 + \cot^2(x) = \csc^2(x)$

Reciprocal Identities

These identities express trigonometric functions as reciprocals of each other.

Function	Reciprocal
$\sin(x)$	$\frac{1}{\csc(x)}$
$\cos(x)$	$\frac{1}{\sec(x)}$
$\tan(x)$	$\frac{1}{\cot(x)}$
$\csc(x)$	$\frac{1}{\sin(x)}$
$\sec(x)$	$\frac{1}{\cos(x)}$
$\cot(x)$	$\frac{1}{\tan(x)}$

Double Angle Identities

Formulas for Double Angles

Double angle identities are used to simplify expressions involving trigonometric functions of $2x$.

• $\sin(2x) = 2\sin(x)\cos(x)$

• $\cos(2x) = \cos^2(x) - \sin^2(x)$

• $\cos(2x) = 2\cos^2(x) - 1$

• $\cos(2x) = 1 - 2\sin^2(x)$

Equation of a Line

Point-Slope and Slope-Intercept Forms

The equation of a tangent line to a function at a given point can be found using the derivative.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $m$ is the slope at $(x_1, y_1)$

• Slope-Intercept Form: $y = mx + b$

• Application: Use the derivative to find the slope at a point, then plug into the point-slope formula.

• Example: For $f(x) = x^2$ at $x = 1$, slope $m = 2$, so $y - 1 = 2(x - 1)$

Practice Problems (Summary)

Types of Problems

The study guide suggests practice problems involving:

• Finding derivatives of functions

• Using the tangent line equation

• Applying the point-slope formula

• Solving for the slope and y-intercept

Additional info: The guide references finding the equation of a tangent line and using derivatives for motion problems, which are standard applications in Calculus I.