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Essential Concepts

Defining the Derivative

• The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment [latex]h[/latex].
• The derivative of a function [latex]f(x)[/latex] at a value [latex]a[/latex] is found using either of the definitions for the slope of the tangent line.
• Velocity is the rate of change of position. As such, the velocity [latex]v(t)[/latex] at time [latex]t[/latex] is the derivative of the position [latex]s(t)[/latex] at time [latex]t[/latex].
  • Average velocity is given by
    [latex]v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}[/latex]
  • Instantaneous velocity is given by
    [latex]v(a)=s^{\prime}(a)=\underset{t\to a}{\lim}\dfrac{s(t)-s(a)}{t-a}[/latex]
• We may estimate a derivative by using a table of values.

The Derivative as a Function

• The derivative of a function [latex]f(x)[/latex] is the function whose value at [latex]x[/latex] is [latex]f^{\prime}(x)[/latex].
• The graph of a derivative of a function [latex]f(x)[/latex] is related to the graph of [latex]f(x)[/latex]. Where [latex]f(x)[/latex] has a tangent line with positive slope, [latex]f^{\prime}(x)>0[/latex]. Where [latex]f(x)[/latex] has a tangent line with negative slope, [latex]f^{\prime}(x)<0[/latex]. Where [latex]f(x)[/latex] has a horizontal tangent line, [latex]f^{\prime}(x)=0[/latex].
• If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
• Higher-order derivatives are derivatives of derivatives, from the second derivative to the [latex]n\text{th}[/latex] derivative.

Differentiation Rules

• The derivative of a constant function is zero.
• The derivative of a power function is a function in which the power on [latex]x[/latex] becomes the coefficient of the term and the power on [latex]x[/latex] in the derivative decreases by 1.
• The derivative of a constant [latex]c[/latex] multiplied by a function [latex]f[/latex] is the same as the constant multiplied by the derivative.
• The derivative of the sum of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the sum of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex].
• The derivative of the difference of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the difference of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex].
• The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
• The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.
• We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.

Derivatives as Rates of Change

• Using [latex]f(a+h)\approx f(a)+f^{\prime}(a)h[/latex], it is possible to estimate [latex]f(a+h)[/latex] given [latex]f^{\prime}(a)[/latex] and [latex]f(a)[/latex].
• The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.
• The population growth rate and the present population can be used to predict the size of a future population.
• Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.

Key Equations

• Difference quotient
  [latex]Q=\dfrac{f(x)-f(a)}{x-a}[/latex]
• Difference quotient with increment [latex]h[/latex]
  [latex]Q=\dfrac{f(a+h)-f(a)}{a+h-a}=\dfrac{f(a+h)-f(a)}{h}[/latex]
• Slope of tangent line
  [latex]m_{\tan}=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}[/latex]
  [latex]m_{\tan}=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}[/latex]
• Derivative of [latex]f(x)[/latex] at [latex]a[/latex]
  [latex]f^{\prime}(a)=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}[/latex]
  [latex]f^{\prime}(a)=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}[/latex]
• Average velocity
  [latex]v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}[/latex]
• Instantaneous velocity
  [latex]v(a)=s^{\prime}(a)=\underset{t\to a}{\lim}\dfrac{s(t)-s(a)}{t-a}[/latex]
• The derivative function
  [latex]f^{\prime}(x)=\underset{h\to 0}{\lim}\dfrac{f(x+h)-f(x)}{h}[/latex]

Glossary

acceleration

is the rate of change of the velocity, that is, the derivative of velocity

amount of change

the amount of a function [latex]f(x)[/latex] over an interval [latex][x,x+h][/latex] is [latex]f(x+h)-f(x)[/latex]

average rate of change

is a function [latex]f(x)[/latex] over an interval [latex][x,x+h][/latex] is [latex]\dfrac{f(x+h)-f(a)}{b-a}[/latex]

constant multiple rule

the derivative of a constant [latex]c[/latex] multiplied by a function [latex]f[/latex] is the same as the constant multiplied by the derivative: [latex]\frac{d}{dx}(cf(x))=cf^{\prime}(x)[/latex]

constant rule

the derivative of a constant function is zero: [latex]\frac{d}{dx}(c)=0[/latex], where [latex]c[/latex] is a constant

derivative

the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

derivative function

gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

difference quotient

of a function [latex]f(x)[/latex] at [latex]a[/latex] is given by

[latex]\dfrac{f(a+h)-f(a)}{h}[/latex] or [latex]\dfrac{f(x)-f(a)}{x-a}[/latex]

difference rule

the derivative of the difference of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the difference of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex]: [latex]\frac{d}{dx}(f(x)-g(x))=f^{\prime}(x)-g^{\prime}(x)[/latex]

differentiable at [latex]a[/latex]

a function for which [latex]f^{\prime}(a)[/latex] exists is differentiable at [latex]a[/latex]

differentiable on [latex]S[/latex]

a function for which [latex]f^{\prime}(x)[/latex] exists for each [latex]x[/latex] in the open set [latex]S[/latex] is differentiable on [latex]S[/latex]

differentiable function

a function for which [latex]f^{\prime}(x)[/latex] exists is a differentiable function

differentiation

the process of taking a derivative

higher-order derivative

a derivative of a derivative, from the second derivative to the [latex]n[/latex]th derivative, is called a higher-order derivative

instantaneous rate of change

the rate of change of a function at any point along the function [latex]a[/latex], also called [latex]f^{\prime}(a)[/latex], or the derivative of the function at [latex]a[/latex]

marginal cost

is the derivative of the cost function, or the approximate cost of producing one more item

marginal revenue

is the derivative of the revenue function, or the approximate revenue obtained by selling one more item

marginal profit

is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item

population growth rate

is the derivative of the population with respect to time

power rule

the derivative of a power function is a function in which the power on [latex]x[/latex] becomes the coefficient of the term and the power on [latex]x[/latex] in the derivative decreases by 1: If [latex]n[/latex] is an integer, then [latex]\frac{d}{dx}(x^n)=nx^{n-1}[/latex]

product rule

the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: [latex]\frac{d}{dx}(f(x)g(x))=f^{\prime}(x)g(x)+g^{\prime}(x)f(x)[/latex]

quotient rule

the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: [latex]\frac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f^{\prime}(x)g(x)-g^{\prime}(x)f(x)}{(g(x))^2}[/latex]

speed

is the absolute value of velocity, that is, [latex]|v(t)|[/latex] is the speed of an object at time [latex]t[/latex] whose velocity is given by [latex]v(t)[/latex]

sum rule

the derivative of the sum of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the sum of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex]: [latex]\frac{d}{dx}(f(x)+g(x))=f^{\prime}(x)+g^{\prime}(x)[/latex]

Study Tips

Tangent Lines

• Practice calculating difference quotients for various functions.
• Visualize the transition from secant lines to the tangent line.
• Remember that the tangent line represents the instantaneous rate of change.
• When finding equations of tangent lines, always identify a point and calculate the slope.

The Derivative of a Function at a Point

• Practice computing derivatives using both definitions.
• Create tables of difference quotients to estimate derivatives.
• Relate the concept of derivative to the slope of the tangent line.
• Remember that the derivative represents an instantaneous rate of change.

Velocities and Rates of Change

• Use tables of values to estimate instantaneous velocities.
• Compare and contrast average velocity and instantaneous velocity.
• Remember that instantaneous velocity is just one example of an instantaneous rate of change.

Derivative Functions

• Familiarize yourself with different notations for derivatives.
• Remember common algebraic techniques like factoring and using conjugates.
• Visualize the connection between the derivative at a point and the derivative function.
• Practice interpreting the meaning of differentiability in terms of the function’s graph.

Graphing a Derivative

• Practice sketching [latex]f'(x)[/latex] given the graph of [latex]f(x)[/latex] and vice versa.
• Pay attention to where [latex]f(x)[/latex] is increasing, decreasing, or has horizontal tangents.
• Look for points where the concavity of [latex]f(x)[/latex] changes.
• Remember that the [latex]y[/latex]-value of [latex]f'(x)[/latex] represents the slope of [latex]f(x)[/latex] at that [latex]x[/latex]-value.

Derivatives and Continuity

• A function can be continuous at a point but fail to be differentiable there.
• Graphically, non-differentiable points often appear as “corners,” “cusps,” or “jumps” in the function.
• When joining functions, ensure both continuity and matching derivatives at the transition point for smoothness.
• The limit definition of the derivative can be used to check differentiability when other methods are unclear.
•  

Higher-Order Derivatives

• Familiarize yourself with different notations for higher-order derivatives.
• Remember that not all functions have derivatives of all orders.
• Look for patterns in successive derivatives of common functions.

The Basic Rules

• Practice applying these rules to various functions, starting with simple ones and progressing to more complex combinations.
• Remember that the Constant Rule means horizontal lines have a slope of zero.
• When using the Power Rule, the exponent becomes the coefficient, and the new exponent decreases by [latex]1[/latex].
• For the Sum and Difference Rules, differentiate each term separately and then combine.
• The Constant Multiple Rule allows you to factor out constants before differentiating.

The Advanced Rules

• Practice identifying when to use each rule based on the function’s structure.
• Remember the order of terms in the quotient rule: “derivative of top times bottom minus derivative of bottom times top, all over bottom squared.”
• For functions with negative exponents, consider rewriting them before applying the extended power rule.
• Look for opportunities to simplify before and after applying any of the rules.
• When using the product rule, the order of terms doesn’t matter due to the commutative property of addition.

Combining Differentiation Rules

• Practice identifying which rules to apply based on the function’s structure
• When dealing with rational functions, apply the quotient rule first, then handle the numerator and denominator separately
• For problems involving horizontal tangent lines, set the derivative equal to zero and solve

Amount of Change Formula

• Practice estimating function values for small [latex]h[/latex] values.
• Compare estimates to actual function values to understand accuracy
• Remember that larger [latex]h[/latex] values generally lead to less accurate estimates

Rate of Change Applications

• Practice interpreting derivatives in context:
  • For motion problems, remember that velocity is the derivative of position, and acceleration is the derivative of velocity.
  • In economics, think of marginal functions as the rate of change of the corresponding total function.
• Sketch graphs to visualize problems:
  • For motion problems, plot position vs. time and velocity vs. time graphs.
  • For economic problems, draw cost, revenue, and profit curves.
• Develop a systematic approach to word problems:
  • Identify the variable(s) and what they represent.
  • Determine which function you’re working with (position, population, cost, etc.).
  • Decide what information the problem is asking for (rate of change, maximum/minimum, etc.).