Calculus I: Differentiation Techniques and Applications Study Guide

Calculus I: Differentiation Techniques and Applications

1. Calculating Derivatives

This section covers the computation of derivatives for various types of functions, including polynomials, trigonometric, exponential, logarithmic, and composite functions. Mastery of differentiation rules is essential for solving these problems.

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

• Product Rule: For $y = u(x)v(x)$, the derivative is $y' = u'(x)v(x) + u(x)v'(x)$.

• Quotient Rule: For $y = \frac{u(x)}{v(x)}$, the derivative is $y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.

• Chain Rule: For $y = f(g(x))$, the derivative is $y' = f'(g(x)) \cdot g'(x)$.

• Trigonometric Derivatives:

  • $\frac{d}{dx}(\sin x) = \cos x$

  • $\frac{d}{dx}(\cos x) = -\sin x$

  • $\frac{d}{dx}(\tan x) = \sec^2 x$

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

• Exponential and Logarithmic Derivatives:

  • $\frac{d}{dx}(e^{x}) = e^{x}$

  • $\frac{d}{dx}(a^{x}) = a^{x} \ln a$

  • $\frac{d}{dx}(\ln x) = \frac{1}{x}$

• Examples:

  • For $y = x^3 \cos x$, use the product rule.

  • For $y = \frac{x}{\sin x}$, use the quotient rule.

  • For $y = e^{x} \sin x$, use the product rule.

  • For $y = \tan^2 \theta$, use the chain rule.

  • For $y = (5x^2 + 4x^{-1})^7$, use the chain rule.

  • For $y = \sin(5x^2 + 4)$, use the chain rule.

  • For $y = 4\sin^{-1}(x)$, recall $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$.

  • For $y = \arctan(5x)$, use the chain rule.

  • For $y = \ln(5x^2 + 4)$, use the chain rule.

  • For $y = x e^{x}$ or $y = y \sin x$, use product rule.

  • For $y = \ln \left( \frac{x}{\sqrt{x+1}} \right)$, use logarithmic properties and chain rule.

2. Second Derivative

The second derivative measures the rate of change of the rate of change, providing information about concavity and inflection points.

• Definition: The second derivative of $f(x)$ is $f''(x) = \frac{d^2 f}{dx^2}$.

• Application: Used to determine concavity and points of inflection.

• Example: For $f(x) = x^2$, $f''(x) = 2$ (constant concavity).

3. Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers, especially when the function is complicated.

• Steps:

  1. Take the natural logarithm of both sides: $\ln y = \ln f(x)$.

  2. Differentiate both sides with respect to $x$.

  3. Solve for $\frac{dy}{dx}$.

• Example: For $y = \ln \left( \frac{x}{\sqrt{x+1}} \right)$, use properties of logarithms to simplify before differentiating.

• Example: For $y = (x^3 + 1)^5$, take $\ln y = 5 \ln(x^3 + 1)$, then differentiate.

4. Marginal Cost and Applications

Marginal cost is the derivative of the cost function with respect to the number of items produced. It estimates the cost of producing one additional item.

• Definition: If $C(x)$ is the cost to produce $x$ items, marginal cost is $C'(x)$.

• Application: Use $C'(x)$ to estimate the change in cost for small changes in production.

• Example: For $C(x) = -0.001x^2 + 0.05x + 0.07x + 800$, compute $C'(x)$ and evaluate at $x = 100$.

5. Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using implicit differentiation.

• Steps:

  1. Identify all variables and rates involved.

  2. Write an equation relating the variables.

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

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

• Example: Two planes moving at right angles: Use the Pythagorean theorem to relate distances, then differentiate to find the rate at which the distance between them changes.

• Example: Water draining from a cylindrical tank: Use the formula for volume $V = \pi r^2 h$ and relate $\frac{dh}{dt}$ to $\frac{dV}{dt}$.

6. Linearization

Linearization approximates a function near a point using the tangent line. It is useful for estimating values of functions for inputs close to the point of tangency.

• Formula: The linearization of $f(x)$ at $x = a$ is $L(x) = f(a) + f'(a)(x - a)$.

• Application: Used to approximate $f(x)$ for $x$ near $a$.

• Example: For $f(x) = \cos(2x)$ at $x = \frac{\pi}{6}$, compute $f(a)$ and $f'(a)$, then write $L(x)$.

7. Summary Table: Differentiation Rules

The following table summarizes key differentiation rules for common functions.

Function	Derivative	Rule Used
$x^n$	$nx^{n-1}$	Power Rule
$e^{x}$	$e^{x}$	Exponential Rule
$\sin x$	$\cos x$	Trigonometric Rule
$\cos x$	$-\sin x$	Trigonometric Rule
$\ln x$	$\frac{1}{x}$	Logarithmic Rule
$f(g(x))$	$f'(g(x))g'(x)$	Chain Rule
$u(x)v(x)$	$u'(x)v(x) + u(x)v'(x)$	Product Rule
$\frac{u(x)}{v(x)}$	$\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$	Quotient Rule

Additional info: The study guide covers differentiation techniques, applications to marginal cost and related rates, and linearization, all central topics in a first-semester college calculus course.