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 the growth rate is zero: $f'(x) = 0$.

• Example: If $f(x) = x^2$ and $g(x) = \sin x$, then $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$.

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 results in cyclic patterns.

• 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 common forms.

• 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 $y = \sin(x^2)$, $\frac{dy}{dx} = \cos(x^2) \cdot 2x$

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: The tangent line at $(x_0, y_0)$ is $y - y_0 = m(x - x_0)$, where $m = \frac{dy}{dx}$ at that point.

• Second Derivative: To find $\frac{d^2y}{dx^2}$, differentiate $\frac{dy}{dx}$ implicitly.

• 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.

• Method: Identify the relationship between variables, differentiate with respect to time $t$, and solve for the desired rate.

• Example: If $x^2 + y^2 = r^2$, then $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$

Applications of the Derivative

Extreme Values and Critical Points (Section 4.1)

Derivatives are used to identify extreme values and critical points of functions, which are important in optimization.

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ does not exist.

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

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

• Identifying Extremes: Use the first derivative test and evaluate endpoints for absolute extremes.

• Example: For $f(x) = x^2$, the minimum occurs at $x = 0$.

Rule	Formula	Example
Power Rule	$\frac{d}{dx}[x^n] = n x^{n-1}$	$\frac{d}{dx}[x^3] = 3x^2$
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$	$\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$
Quotient Rule	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx}\left[\frac{x}{\sin x}\right] = \frac{1 \cdot \sin x - x \cdot \cos x}{\sin^2 x}$
Chain Rule	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x$
Implicit Differentiation	$\frac{dy}{dx}$ from implicit equation	$x^2 + y^2 = 1 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

Additional info: The syllabus skips Section 3.6 and focuses on derivative rules, applications, and related rates, which are central to calculus. The table summarizes key differentiation rules and examples for quick reference.