📐Analytic Geometry and Calculus Unit 4 – Differentiation Rules and Their Applications

Key Concepts and Definitions

• Differentiation calculates the rate of change of a function at a given point
• Derivative represents the instantaneous rate of change or slope of the tangent line at a point
• Notation for derivatives includes $\frac{d}{dx}f(x)$, $f'(x)$, or $\frac{dy}{dx}$
  • $\frac{d}{dx}$ is the derivative operator, indicating the operation of differentiation with respect to $x$
  • $f'(x)$ is the prime notation, denoting the derivative of the function $f(x)$
  • $\frac{dy}{dx}$ is the Leibniz notation, representing the derivative of $y$ with respect to $x$
• Differentiability a function is differentiable at a point if its derivative exists at that point
  • Differentiability is a stronger condition than continuity
  • If a function is differentiable at a point, it must also be continuous at that point
• Tangent line a straight line that touches a curve at a single point without crossing it
• Normal line a line perpendicular to the tangent line at the point of tangency

Basic Differentiation Rules

• Constant Rule $\frac{d}{dx}(c) = 0$, where $c$ is a constant
• Power Rule $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a real number
• Constant Multiple Rule $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}f(x)$, where $c$ is a constant
• Sum Rule $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$
  • The derivative of a sum is the sum of the derivatives
• Difference Rule $\frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}f(x) - \frac{d}{dx}g(x)$
  • The derivative of a difference is the difference of the derivatives
• Exponential Function Rule $\frac{d}{dx}(e^x) = e^x$
• Natural Logarithm Rule $\frac{d}{dx}(\ln x) = \frac{1}{x}$, for $x > 0$

Chain Rule and Its Applications

• Chain Rule $\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$
  • If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
  • The Chain Rule allows for the differentiation of composite functions
• Composition of Functions $f(g(x))$ is a composite function, where the output of $g(x)$ is the input of $f$
• General Power Rule $\frac{d}{dx}(u(x))^n = n(u(x))^{n-1} \cdot \frac{d}{dx}u(x)$, where $u$ is a function of $x$
• Generalized Exponential Rule $\frac{d}{dx}(a^{u(x)}) = a^{u(x)} \cdot \ln(a) \cdot \frac{d}{dx}u(x)$, where $a > 0$ and $a \neq 1$
• Logarithmic Differentiation a technique used to differentiate functions of the form $f(x) = (u(x))^{v(x)}$
  • Take the natural logarithm of both sides, then use properties of logarithms and the Chain Rule to differentiate

Product and Quotient Rules

• Product Rule $\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$
  • The derivative of a product is the first function times the derivative of the second, plus the second function times the derivative of the first
• Quotient Rule $\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{(g(x))^2}$, where $g(x) \neq 0$
  • The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator
• Reciprocal Rule $\frac{d}{dx}(\frac{1}{f(x)}) = -\frac{f'(x)}{(f(x))^2}$, a special case of the Quotient Rule
• Combining Rules often need to apply multiple differentiation rules to find the derivative of a complex function
  • Identify the overall structure of the function (product, quotient, composite)
  • Apply the appropriate rule or combination of rules to differentiate each part

Implicit Differentiation

• Implicit Function a function defined by an equation in which the dependent variable is not explicitly expressed in terms of the independent variable
  • Example $x^2 + y^2 = 25$ defines a circle, but $y$ is not explicitly written as a function of $x$
• Implicit Differentiation a method for finding the derivative of an implicit function
  • Differentiate both sides of the equation with respect to the independent variable (usually $x$)
  • Treat the dependent variable (usually $y$) as a function of $x$ and apply the Chain Rule when necessary
  • Solve the resulting equation for $\frac{dy}{dx}$
• Tangent Line to Implicit Curve can find the equation of the tangent line at a point by using implicit differentiation to find $\frac{dy}{dx}$
• Related Rates problems involve finding the rate of change of one quantity in terms of the rate of change of another quantity
  • Write an equation relating the quantities
  • Differentiate implicitly with respect to time
  • Substitute known values and solve for the desired rate

Higher-Order Derivatives

• Higher-Order Derivatives derivatives of derivatives
  • The second derivative is the derivative of the first derivative
  • The third derivative is the derivative of the second derivative, and so on
• Notation for higher-order derivatives $f''(x)$, $f'''(x)$, $f^{(n)}(x)$, or $\frac{d^2y}{dx^2}$, $\frac{d^3y}{dx^3}$, $\frac{d^ny}{dx^n}$
• Physical Interpretations
  • First derivative represents the rate of change (velocity)
  • Second derivative represents the rate of change of the rate of change (acceleration)
• Concavity and Inflection Points
  • Concavity determines whether a curve is concave up (opens upward) or concave down (opens downward)
    • If $f''(x) > 0$, the graph is concave up
    • If $f''(x) < 0$, the graph is concave down
  • Inflection Point a point where the concavity changes
    • At an inflection point, $f''(x) = 0$ or $f''(x)$ does not exist

Applications in Real-World Problems

• Optimization finding the maximum or minimum value of a function subject to certain constraints
  • Determine the function to be optimized (objective function)
  • Identify the constraints and express them as equations or inequalities
  • Find the critical points by setting the derivative of the objective function equal to zero
  • Evaluate the objective function at the critical points and endpoints (if applicable) to determine the maximum or minimum value
• Marginal Analysis studying the effect of a small change in one variable on another variable
  • Marginal Cost the change in total cost resulting from producing one additional unit
  • Marginal Revenue the change in total revenue resulting from selling one additional unit
  • Marginal Profit the change in total profit resulting from producing and selling one additional unit
• Growth and Decay Models
  • Exponential Growth $\frac{dP}{dt} = kP$, where $P$ is the population size and $k$ is the growth rate
  • Exponential Decay $\frac{dQ}{dt} = -kQ$, where $Q$ is the quantity and $k$ is the decay rate
• Motion Along a Line
  • Position $s(t)$ the distance from the origin at time $t$
  • Velocity $v(t) = \frac{ds}{dt}$ the rate of change of position with respect to time
  • Acceleration $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ the rate of change of velocity with respect to time

Common Mistakes and How to Avoid Them

• Forgetting to apply the Chain Rule when differentiating composite functions
  • Identify the "inner" and "outer" functions
  • Differentiate the outer function, then multiply by the derivative of the inner function
• Misapplying the Product or Quotient Rule
  • Be careful with the order of the terms and the signs
  • Remember to differentiate both the first and second functions in the product or quotient
• Incorrectly handling constants or coefficients
  • Constants are differentiated to zero
  • Coefficients are factored out before differentiating
• Losing sight of the variable with respect to which you are differentiating
  • Pay attention to the independent variable (usually $x$ or $t$)
  • Be consistent in your notation
• Failing to simplify or combine like terms after differentiating
  • Simplify the derivative by combining like terms
  • Factor out common terms to make the expression more concise
• Not checking for continuity or differentiability
  • Ensure the function is continuous at the point of interest
  • Check for any points where the derivative may not exist (corners, cusps, or vertical tangents)
• Misinterpreting the results or units
  • Be aware of the context and the units involved
  • Interpret the derivative in terms of the original problem or application