Calculus I: Differentiation and Applications Study Notes

Differentiation: Rules and Applications

Finding Derivatives of Functions

Calculus often begins with the study of derivatives, which measure the rate at which a function changes. The derivative of a function y = f(x) at a point x gives the slope of the tangent line to the curve at that point.

• Definition: The derivative of f(x) with respect to x is denoted as f'(x) or \( \frac{dy}{dx} \).

• Basic Rules:

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

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

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

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

  • Chain Rule: \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)

• Example: For \( y = 1 + \sqrt{x + x^2} \), use the chain rule and power rule to differentiate.

• Example: For \( y = \sec^2(x) \tan(x^2) \), use the product rule and chain rule.

Second Derivatives

The second derivative of a function, denoted as \( \frac{d^2y}{dx^2} \), measures the rate of change of the rate of change, or the concavity of the function.

• Example: Find the second derivative of \( y = \ln(x^2) \).

• Application: Second derivatives are used to determine points of inflection and the concavity of graphs.

Implicit Differentiation

When a function is not given explicitly as y = f(x), but rather as an equation involving both x and y, implicit differentiation is used.

• Example: For \( x^2y = y^3 + 3 \), find \( \frac{dy}{dx} \) at the point (2, 1).

• Steps:

  1. Differentiate both sides with respect to x, treating y as a function of x.

  2. Solve for \( \frac{dy}{dx} \).

Derivatives of Trigonometric and Inverse Trigonometric Functions

Trigonometric and inverse trigonometric functions have specific derivative formulas.

• Key Formulas:

  • \( \frac{d}{dx} \sin(x) = \cos(x) \)

  • \( \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 - x^2}} \)

  • \( \frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2} \)

• Example: Find \( \frac{dy}{dx} \) for \( y = \arctan(2^x) \).

• Example: Find \( \frac{dy}{dx} \) for \( y = \arcsin(x) + \arcsin(\sqrt{1 - x^2}) \), \( x > 0 \).

Slopes and Tangent Lines

The slope of a curve at a point is given by the derivative at that point. The equation of the tangent line can be found using the point-slope form:

• Formula: \( y - y_1 = m(x - x_1) \), where m is the slope at \( (x_1, y_1) \).

• Example: Find the slope of the curve \( x^2y = y^3 + 3 \) at (2, 1).

• Example: Find the equation of the tangent line to \( x^n \) at \( x = c \).

Related Rates and Motion Along a Line

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using the chain rule. Motion problems use derivatives to find velocity and acceleration.

• Example: If the position of a body moving along the x-axis is \( s = 3 - 2t + t^2 \), find when the body changes direction and its velocity and acceleration at that time.

• Key Formulas:

  • Velocity: \( v(t) = \frac{ds}{dt} \)

  • Acceleration: \( a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} \)

Perpendicular Tangents

To find points where the tangent to a curve is perpendicular to a given line, use the negative reciprocal of the slope of the given line.

• Example: Find all points (x, y) on the graph of \( y = \sqrt{x - 4} \) where the tangent is perpendicular to the line \( y = 4x + 3 \).

• Method: Set the derivative equal to the negative reciprocal of the given line's slope and solve for x.

Table: Derivatives of Common Functions

The following table summarizes the derivatives of some common functions:

Function	Derivative
\( x^n \)	\( n x^{n-1} \)
\( \sin(x) \)	\( \cos(x) \)
\( \cos(x) \)	\( -\sin(x) \)
\( \tan(x) \)	\( \sec^2(x) \)
\( \arcsin(x) \)	\( \frac{1}{\sqrt{1 - x^2}} \)
\( \arctan(x) \)	\( \frac{1}{1 + x^2} \)
\( \ln(x) \)	\( \frac{1}{x} \)

Additional info:

• Some questions require using the chain rule, product rule, and implicit differentiation.

• Applications include finding tangent lines, related rates, and points of perpendicularity.