Calculus Study Guide: Derivatives and Applications

Derivatives and Their Rules

Basic Derivative Rules (Section 3.3)

The derivative measures the instantaneous rate of change of a function. Calculating derivatives is fundamental in calculus, and several rules simplify the process.

• Power Rule: For $f(x) = x^n$, the derivative is $f'(x) = n x^{n-1}$.

• Sum Rule: The derivative of a sum is the sum of the derivatives: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$.

• Constant Multiple Rule: $\frac{d}{dx}[c f(x)] = c f'(x)$, where $c$ is a constant.

• Finding Tangent Slopes: To find where the tangent has a specific slope $m$, solve $f'(x) = m$ for $x$.

• Higher-Order Derivatives: The second derivative $f''(x)$ measures the rate of change of the first derivative; higher-order derivatives follow similarly.

• Example: For $f(x) = x^3$, $f'(x) = 3x^2$, $f''(x) = 6x$.

Product and Quotient Rules (Section 3.4)

When differentiating products or quotients of functions, specialized rules are used.

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

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

• Instantaneous Growth Rate: The derivative $f'(x)$ gives the instantaneous rate of change at $x$.

• Steady-State Population: In population models, steady-state occurs where $f'(x) = 0$.

• Example: For $f(x) = x^2 \cdot \sin(x)$, $f'(x) = 2x \sin(x) + x^2 \cos(x)$.

Derivatives of Trigonometric Functions (Section 3.5)

Trigonometric functions have well-defined derivatives, which are essential in many calculus problems.

• Basic Derivatives:

  • $\frac{d}{dx}[\sin(x)] = \cos(x)$

  • $\frac{d}{dx}[\cos(x)] = -\sin(x)$

  • $\frac{d}{dx}[\tan(x)] = \sec^2(x)$

• Higher-Order Derivatives: Repeated differentiation of trigonometric functions often cycles through the original function and its negatives.

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

Advanced Differentiation Techniques

Chain Rule (Section 3.7)

The chain rule is used to differentiate composite functions. There are two main forms of the chain rule.

• Standard Form: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

• Theorem 3.12 (Alternative Form): If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

• Example: For $f(x) = \sin(x^2)$, $f'(x) = \cos(x^2) \cdot 2x$

Implicit Differentiation (Section 3.8)

Implicit differentiation is used when functions are defined implicitly rather than explicitly.

• Implicit Differentiation: Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

• Tangent Line Equation: At point $(x_0, y_0)$, the tangent line is $y - y_0 = m(x - x_0)$, where $m = \frac{dy}{dx}$ at $(x_0, y_0)$.

• Second Derivative: After finding $\frac{dy}{dx}$, differentiate again to find $\frac{d^2y}{dx^2}$, using implicit differentiation.

• Example: For $x^2 + y^2 = 1$, $\frac{dy}{dx} = -\frac{x}{y}$

Related Rates (Section 3.9)

Related rates problems involve finding the rate at which one quantity changes in relation to another.

• Procedure:

  1. Identify all variables and their rates of change.

  2. Write an equation relating the variables.

  3. Differentiate both sides with respect to time $t$.

  4. Substitute known values and solve for the unknown rate.

• Example: If a circle's radius increases at 2 cm/s, the area increases at $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$.

Applications of the Derivative

Extreme Values and Critical Points (Section 4.1)

Derivatives are used to identify extreme values (maximums and minimums) and critical points of functions.

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ is undefined.

• Absolute Extreme Values: The highest or lowest value of $f(x)$ on a given interval.

• Local Extreme Values: Maximum or minimum values in a neighborhood around a point.

• Procedure:

  1. Find $f'(x)$ and solve $f'(x) = 0$ for critical points.

  2. Evaluate $f(x)$ at critical points and endpoints (if interval is closed).

  3. Compare values to determine absolute and local extremes.

• Example: For $f(x) = x^2 - 4x + 3$, critical point at $x = 2$, $f(2) = -1$.