Elementary Derivative Rules

Introduction to Elementary Derivative Rules

Motivating Questions for Derivative Rules

• What are the alternate notations for the derivative?

• How can the algebraic structure of a function be used to compute a formula for $f'(x)$?

• What is the derivative of a power function of the form $f(x) = x^n$?

• What is the derivative of an exponential function of the form $f(x) = a^x$?

• If the derivative of $y = f(x)$ is known, what is the derivative of $y = kf(x)$, where $k$ is a constant?

• If the derivatives of $y = f(x)$ and $y = g(x)$ are known, how is the derivative of $y = f(x) + g(x)$ computed?

Chapter 1 Review: The Concept of the Derivative

• The derivative $f'(x)$ of a function $f(x)$ measures the instantaneous rate of change of $f$ with respect to $x$.

• The derivative also represents the slope of the tangent line to $y = f(x)$ at any given value of $x$.

• Previously, the focus was on interpreting the derivative graphically or as a meaningful rate of change in physical contexts.

• To calculate the derivative at a specific point, the limit definition was used:
  $f'(x) = rac{ ext{lim}}{h o 0} rac{f(x + h) - f(x)}{h}$

• Goal of Chapter 2: Investigate patterns and rules derived from the limit definition to find derivative formulas quickly, without directly using the limit definition. For example, applying shortcuts to differentiate functions like $g(x) = 4x^7 - ext{sin}(x) + 3e^x$ by observation.

Key Notations for the Derivative

Beyond $f'(x)$, several other notations are commonly used:

• Leibniz Notation: When considering the relationship between $y$ and $x$, the derivative of $y$ with respect to $x$ is denoted by $rac{dy}{dx}$.

  • Example: If $y = x^2$, then $rac{dy}{dx} = 2x$.

  • Origin: This notation relates to the slope of a line, measured by $rac{ ext{Δ}y}{ ext{Δ}x}$. However, $rac{dy}{dx}$ is viewed as a single symbol, not a quotient of two quantities, even though it's read as "change in $y$ over change in $x$" or "dee-y dee-x."

• Operator Notation: The instruction "take the derivative of the quantity in $ext{□}$ with respect to $x$" is written as $rac{d}{dx}[ ext{□}]$.

  • Example: $rac{d}{dx}[x^2] = 2x$.

• Variable Flexibility: The independent variable can vary.

  • If $f(z) = z^2$, then $f'(z) = 2z$.

  • If $y = t^2$, then $rac{dy}{dt} = 2t$.

• Second Derivatives: Notations extend to higher-order derivatives.

  • Example: f''(z) = rac{d}{dz}igg[ rac{df}{dz}igg] = rac{d^2f}{dz^2}.

Basic Derivative Rules: Constant, Power, and Exponential Functions

The Derivative of a Constant Function

• Rule: For any real number $c$, if $f(x) = c$, then $f'(x) = 0$.

• Explanation: The graph of a constant function is a horizontal line, which has a slope of zero at every point.

• Examples:

  • If $f(x) = 7$, then $f'(x) = 0$.

  • $rac{d}{dx}[ ext{√}3] = 0$.

The Derivative of a Power Function

• Preview Activity 2.1.1 Exploration:

  • Using the limit definition to find derivatives for $f(x) = x^2$, $f(x) = x^3$, and $f(x) = x^4$ (using binomial expansion for $(x+h)^4$) leads to a conjecture.

  • Conjectures for $f(x) = x^5$ and $f(x) = x^{13}$ suggest a pattern.

  • General conjecture: for $f(x) = x^n$, where $n$ is a positive integer, $f'(x) = nx^{n-1}$.

• Rule: For any nonzero real number $n$ (positive or negative), if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

  • This rule can be formally proven for any nonzero real number.

• Examples:

  • If $g(z) = z^{-3}$, then $g'(z) = -3z^{-4}$.

  • If $h(t) = t^{7/5}$, then $rac{dh}{dt} = rac{7}{5}t^{2/5}$.

  • $rac{d}{dq}[q^ ext{π}] = ext{π}q^{ ext{π}-1}$.

The Derivative of an Exponential Function

• Rule: For any positive real number $a$, if $f(x) = a^x$, then $f'(x) = a^x ext{ln}(a)$.

  • (Note: This rule is stated without immediate justification; justification is explored in exercises and later activities.)

• Examples:

  • If $f(x) = 2^x$, then $f'(x) = 2^x ext{ln}(2)$.

  • If $p(t) = 10^t$, then $p'(t) = 10^t ext{ln}(10)$.

• Special Case: The Natural Exponential Function

  • When the base $a = e$, where $e$ is the base of the natural logarithm, the rule becomes:
    $rac{d}{dx}[e^x] = e^x ext{ln}(e) = e^x ext{⋅} 1 = e^x$

  • Significance: The derivative of $f(x) = e^x$ is the function itself ($f'(x) = e^x$). This is a uniquely important property.

• Distinction between Power and Exponential Functions:

  • Power functions: Variable is in the base, e.g., $x^2$, $x^n$.

  • Exponential functions: Variable is in the exponent (power), e.g., $2^x$, $a^x$.

  • The form of their derivatives differs significantly.

Activity 2.1.2: Applying Basic Rules

• Use constant, power, and exponential rules to find derivatives of functions, applying proper notation (e.g., $f'(x)$ or $rac{dh}{dz}$).

  • (a) $f(t) = ext{π}$ (Constant rule)

  • (b) $g(z) = 7^z$ (Exponential rule)

  • (c) $h(w) = w^{3/4}$ (Power rule)

  • (d) $p(x) = 3^{1/2}$ (Constant rule, since $3^{1/2}$ is a constant)

  • (e) $r(t) = ( ext{√}2)^t$ (Exponential rule, $a = ext{√}2$)

  • (f) $s(q) = q^{-1}$ (Power rule, $n=-1$)

  • (g) $m(t) = rac{1}{t^3} = t^{-3}$ (Power rule, $n=-3$)

Combining Derivative Rules: Constant Multiples and Sums

These rules allow differentiating functions constructed from basic functions through algebraic combinations, such as polynomials (e.g., $p(t) = 3t^5 - 7t^4 + t^2 - 9$).

The Constant Multiple Rule

• Rule: For any real number $k$, if $f(x)$ is a differentiable function with derivative $f'(x)$, then $rac{d}{dx}[kf(x)] = kf'(x)$.

• In Words: "The derivative of a constant times a function is the constant times the derivative of the function."

• Graphical Interpretation: Multiplying a function by a constant $k$ vertically stretches its graph by a factor of $|k|$ (and reflects it across the $x$-axis if k < 0). This stretch affects the slope, making the slope of $kf(x)$ exactly $k$ times as steep as the slope of $f(x)$.

• Examples:

  • If $g(t) = 3 ext{⋅} 5^t$, then $g'(t) = 3 ext{⋅} (5^t ext{ln}(5))$.

  • $rac{d}{dz}[5z^{-2}] = 5(-2z^{-3}) = -10z^{-3}$.

The Sum Rule

• Rule: If $f(x)$ and $g(x)$ are differentiable functions with derivatives $f'(x)$ and $g'(x)$ respectively, then $rac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$.

• In Words: "The derivative of a sum is the sum of the derivatives."

• Implication: A sum of two differentiable functions is also differentiable.

• Extension to Differences: Since a difference can be viewed as a sum with a constant multiple (e.g., $f(x) - g(x) = f(x) + (-1 ext{⋅} g(x))$), the Sum Rule and Constant Multiple Rule together imply:
  $rac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$
  or "the derivative of a difference is the difference of the derivatives."

• Examples:

  • $rac{d}{dw}(2^w + w^2) = 2^w ext{ln}(2) + 2w$.

  • If $h(q) = 3q^6 - 4q^{-3}$, then $h'(q) = 3(6q^5) - 4(-3q^{-4}) = 18q^5 + 12q^{-4}$.

Activity 2.1.3: Applying Combined Rules

• Compute derivatives using constant, power, exponential functions rules, along with the Constant Multiple and Sum Rules.

• Important Caveat: Product and Quotient Rules are not yet learned. Algebraic manipulation (e.g., expanding products, dividing terms) may be necessary before applying current rules.

• Examples of functions to differentiate:

  • (a) $f(x) = x^{5/3} - x^4 + 2x$

  • (b) $g(x) = 14e^x + 3x^5 - x$

  • (c) $h(z) = ext{√}z + rac{1}{z^4} + 5z$ (Rewrite as $z^{1/2} + z^{-4} + 5z$)

  • (d) $r(t) = ext{√}5^3 t^7 - ext{π}e^t + e^4$ (Recognize $ext{√}5^3$ and $e^4$ as constants)

  • (e) $s(y) = (y^2 + 1)(y^2 - 1)$ (Expand to $y^4 - 1$ first)

  • (f) $q(x) = rac{x^3 - x + 2}{x}$ (Divide each term by $x$ to get $x^2 - 1 + 2x^{-1}$)

  • (g) $p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12$

Terminology Simplification

• Instead of "take the derivative of $f$", we often say "differentiate $f$".

• If a derivative exists at a point, we say "$f$ is differentiable at that point," or that "$f$ can be differentiated."

General Observations on Derivatives from Rules

• Polynomials: The derivative of any polynomial function will be another polynomial function. The degree of the derivative is one less than the degree of the original function.

  • Example: If $p(t) = 7t^5 - 4t^3 + 8t$ (degree 5), its derivative $p'(t) = 35t^4 - 12t^2 + 8$ (degree 4).

• Exponential Functions: The derivative of any exponential function is another exponential function.

  • Example: If $g(z) = 7 ext{⋅} 2^z$, then $g'(z) = 7 ext{⋅} 2^z ext{ln}(2)$, which is also exponential.

• Core Meaning Endures: All the meaning of the derivative developed in Chapter 1 still holds: it measures the instantaneous rate of change and the slope of the tangent line.

Activity 2.1.4: Applying Derivatives to Questions

• (a) Find the slope of the tangent line to $h(z) = ext{√}z + rac{1}{z}$ at $z = 4$. (Requires calculating $h'(4)$).

• (b) Cell population growth: $P(t) = 2(1.37)^t + 32$, where $t$ is in days.

  • (i) Determine the instantaneous rate of population growth on day 4, including units. (Requires $P'(4)$).

  • (ii) Is the population growing at an increasing or decreasing rate on day 4? Explain. (Requires analyzing the second derivative's sign at $t=4$, or the trend of $P'(t)$).

• (c) Find an equation for the tangent line to the curve $p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12$ at $a = -1$. (Requires $p(-1)$ for the point and $p'(-1)$ for the slope).

• (d) Clarify the difference between finding the slope of the tangent line and the equation of the tangent line.

Summary of Elementary Derivative Rules (2.1.5)

• Notation: For a differentiable function $y = f(x)$, its derivative can be expressed as $f'(x)$, $rac{df}{dx}$, $rac{dy}{dx}$, or $rac{d}{dx}[f(x)]$.

• Limit Definition Leads to Rules: The limit definition reveals patterns for computing derivative formulas directly.

• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$ for any real number $n$ (other than 0).

• Exponential Rule: For any positive real number $a$, if $f(x) = a^x$, then $f'(x) = a^x ext{ln}(a)$.

• Constant Multiple and Sum Rules: If $f(x)$ and $g(x)$ are differentiable functions with derivatives $f'(x)$ and $g'(x)$, and $a$ and $b$ are constants, then:
  $rac{d}{dx}[af(x) + bg(x)] = af'(x) + bg'(x)$

Exercises (2.1.6)

This section contains various exercises to practice applying the constant, power, exponential, constant multiple, and sum/difference rules. These include:

• Finding derivatives of various functions.

• Finding values of $x$ for which $f'(x) = 0$.

• Finding equations for tangent lines.

• Calculating rates of change in word problems.

• Problems involving piecewise linear functions and interpreting derivatives from graphs.

• A detailed problem (Problem 11) exploring the exponential derivative rule $rac{d}{dx}[a^x] = a^x ext{ln}(a)$ using the limit definition, leading to the special case of $e^x$. It shows that $f'(x) = a^x ext{⋅} ext{lim}_{h o 0} rac{a^h - 1}{h}$. It asks to find $a$ such that this limit is 1, which happens when $a=e$.