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: $\left( f(x) + g(x) \right)' = f'(x) + g'(x)$.

• Constant Multiple Rule: $\left( c f(x) \right)' = c f'(x)$, where $c$ is a constant.

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

• 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: $\left( f(x) g(x) \right)' = f'(x) g(x) + f(x) g'(x)$

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

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

• Steady-State Population: Occurs where the growth rate is zero, i.e., $f'(x) = 0$.

• Example: If $P(t) = t^2 e^t$, then $P'(t) = 2t e^t + t^2 e^t$.

Derivatives of Trigonometric Functions (Section 3.5)

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

• 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 follows cyclical 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.

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

• Finding Tangent Lines: Use the derivative to find the slope at a point, then apply the point-slope form: $y - y_0 = m(x - x_0)$.

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

• Example: For $x^2 + y^2 = 1$, $2x + 2y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

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 relationships.

  2. Differentiate with respect to time $t$.

  3. Substitute known values and solve for the unknown 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 maximum and minimum values of functions, both locally and absolutely.

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

• Local Extreme Values: Local maxima and minima occur at critical points.

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

• 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 extrema.

• Example: For $f(x) = x^2 - 4x + 3$, $f'(x) = 2x - 4$, set $2x - 4 = 0 \Rightarrow x = 2$.

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	Differentiate both sides, solve for $\frac{dy}{dx}$	$x^2 + y^2 = 1 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

Additional info: The syllabus skips Section 3.6 and focuses on derivative rules, trigonometric derivatives, chain rule, implicit differentiation, related rates, and applications to extreme values. This guide expands brief syllabus points into full academic explanations and examples for exam preparation.