Study Notes: Differentiation and Applications for College Algebra

Functions and Differentiation

Geometric Interpretation of the Derivative

The derivative of a function at a point provides the slope of the tangent line to the curve at that point. This is a fundamental concept in calculus and is essential for understanding rates of change in algebraic functions.

• Slope of the Tangent Line: For a function $y = f(x)$ at the point $P(a, f(a))$, the slope is given by: $m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Tangent Line Equation: The equation of the tangent line at $x = a$ is: $y - f(a) = m(x - a)$

• Interpretation: The tangent line represents the instantaneous rate of change of the function at a specific point.

Rate of Change

The rate of change describes how a quantity varies with respect to another. In algebra, this is often used to analyze functions and their behavior.

• Average Rate of Change: Over an interval $[a, a + \Delta x]$, the average rate is: $\frac{\Delta y}{\Delta x} = \frac{f(a + \Delta x) - f(a)}{\Delta x}$

• Instantaneous Rate of Change: At $x = a$, the instantaneous rate is: $f'(a) = \lim_{\Delta x \to 0} \frac{f(a + \Delta x) - f(a)}{\Delta x}$

Definition of the Derivative

The derivative of a function $y = f(x)$ at a number $a$ is defined as:

• $f'(a) := \lim_{\Delta x \to 0} \frac{f(a + \Delta x) - f(a)}{\Delta x}$

• Alternative notation: $f'(a)$ (Newton), $\frac{df}{dx}(a)$ (Leibniz)

• The derivative represents both the slope of the tangent and the instantaneous rate of change.

Derivatives of Fundamental Functions

Common derivatives include:

• $(C)' = 0$

• $(x^\alpha)' = \alpha x^{\alpha-1}$

• $(a^x)' = a^x \ln a$

• $(\log_a |x|)' = \frac{1}{x \ln a}$

• $(\sin x)' = \cos x$

• $(\cos x)' = -\sin x$

• $(\tan x)' = 1 + \tan^2 x$

Applications in Economics: Marginal Quantity and Elasticity

Marginal Quantity

In economics, the marginal quantity refers to the additional change in a dependent variable when the independent variable increases by one unit.

• For $y = f(x)$, when $\Delta x$ is small: $\Delta y \approx f'(x_0) \cdot \Delta x$

• If $\Delta x = 1$, then $f(x_0 + 1) - f(x_0) \approx f'(x_0)$

• Marginal function: $M_y(x) = f'(x)$

Elasticity

Elasticity measures the percentage change in the dependent variable for a 1% change in the independent variable.

• Elasticity at $x$ is: $\varepsilon = E_y(x) := \frac{y'(x) x}{y(x)}$

• Interpretation: If $x$ increases by 1%, $y$ changes by approximately $E_y(x)$ percent.

Techniques of Differentiation

General Rules

• $(Cf)' = Cf'$

• $(f \pm g)' = f' \pm g'$

• $(fg)' = f'g + fg'$

• $\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}$ (for $g \neq 0$)

Chain Rule

• For $y = f(u(x))$, $y' = f'(u(x)) \cdot u'(x)$

• Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Implicit Functions and Their Derivatives

Implicit Differentiation

Some functions are defined implicitly by equations involving both $x$ and $y$. To find derivatives, differentiate both sides with respect to $x$, applying the chain rule as needed.

• Example: If $F(x, y) = C$, differentiate to find $y'$.

• To find higher-order derivatives, differentiate the resulting expression again.

Higher-Order Derivatives

Definition and Computation

Higher-order derivatives describe the rate of change of the derivative itself. The $n$-th derivative is denoted $f^{(n)}$ or $\frac{d^n f}{dx^n}$.

• Compute successive derivatives to find patterns.

• Use mathematical induction to prove general formulas.

Linear Approximation and Differentials

Linear Approximation

For $x$ near $a$, a function $f(x)$ can be approximated by its tangent line:

• $f(x) \approx f(a) + f'(a)(x - a)$

• This is useful for estimating values and understanding local behavior.

Differentials

The differential $dy$ approximates the change in $y$ for a small change in $x$:

• $dy(a) = y'(a) \Delta x$

• For $y = x$, $dx(a) = \Delta x$

• Generally, $dy = y' dx$ or $df = f' dx$

• High-order differentials: $d^n y = y^{(n)} dx^n$

Additional info: The image illustrates the geometric relationship between the tangent line, the curve, and the increments $\Delta x$, $\Delta y$, and $dy$.

Applications: L'Hôpital's Rule

Indeterminate Forms and L'Hôpital's Rule

L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

• If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is indeterminate, then: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

• May need to apply the rule multiple times until the limit is determinate.

Extrema of Functions

Definitions

Extrema refer to the maximum and minimum values of a function, either locally or globally.

• Local Maximum: $f(x_0)$ is a local maximum if $f(x) \leq f(x_0)$ for all $x$ near $x_0$.

• Local Minimum: $f(x_0)$ is a local minimum if $f(x) \geq f(x_0)$ for all $x$ near $x_0$.

• Global Maximum: $f(x_0) = \max_{x \in D} f(x)$

• Global Minimum: $f(x_0) = \min_{x \in D} f(x)$

Finding Extrema

• Necessary Condition: If $f$ has a local extremum at $x = c$, then $f'(c) = 0$ (stationary point).

• First Derivative Test:

  • If $f'$ changes from positive to negative at $c$, $f$ has a local maximum.

  • If $f'$ changes from negative to positive at $c$, $f$ has a local minimum.

  • If $f'$ does not change sign, no extremum at $c$.

• Second Derivative Test:

  • If $f'(c) = 0$ and $f''(c) > 0$, $f$ has a local minimum at $c$.

  • If $f'(c) = 0$ and $f''(c) < 0$, $f$ has a local maximum at $c$.

Steps to Find Extrema

• Find stationary points by solving $f'(x) = 0$.

• Apply the first or second derivative test at each stationary point.

• For global extrema on $[a, b]$, compare values at stationary points and endpoints.

Summary Table: Derivative Rules

Rule	Formula
Constant Rule	$(C)' = 0$
Power Rule	$(x^\alpha)' = \alpha x^{\alpha-1}$
Sum/Difference Rule	$(f \pm g)' = f' \pm g'$
Product Rule	$(fg)' = f'g + fg'$
Quotient Rule	$\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}$
Chain Rule	$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$