MA005 Study Guide

Unit 3: Derivatives

3a. Find the derivative of a function f(x) using the definition

  • What is the formal definition of the derivative?
  • How does this formal definition relate to the slope of the tangent line?

The formal definition of the derivative is related to the equation for the slope of a tangent line: it is just the difference quotient in the limit. See Figure 3a.1. It is derived from the slope of the secant line, but as the difference between the two points goes to zero. The derivative produces a function that can be evaluated at any point on the graph to find the slope of the tangent line.

Figure 3a.1.
Figure 3a.1.

To review, see The Definition of a Derivative.

 

3b. Recognize and use the common equivalent notations for the derivative

  • What are three notations for the first derivative?
  • How does each definition emphasize different meanings of differentiation?
  • Define magnification error.

There are three common notations for the first derivative.

  1. The prime notation (f'(x)) emphasizes the relationship to the function f(x) and is a commonly used formula because it requires the fewest additional characters.
  2. The differential notation (d(f)) emphasizes that the derivative is an operation on the function f, but it does not specify which variable the function is in.
  3. The Leibniz notation ( \frac{df}{dx}) emphasizes the relationship to the slope by representing the difference in the two function values divided by the difference in the two x values just as the slope formula.

Magnification factor f'(x) is the magnification factor of the function f for points close to x. If a and b are two points very close to x, then the distance between f(a) and f(b) will be close to f'(x) times the original distance between a and b: f(b) − f(a) ≈ f'(x)(b − a).

To review, see The Definition of a Derivative.

 

3c. State the graph and rate meanings of a derivative

  • What are some common interpretations of the first derivative?
  • Explain why it makes sense that the derivative of a constant function must logically be zero without using the formula?
  • Define horizontal tangent lines.

There are many common interpretations of the derivative depending on the area of application. In general, the derivative is the slope of the tangent line or the instantaneous rate of change. In a physical context, the derivative is the velocity of a particle moving on a path. The derivative of the velocity is the rate of change of the velocity, or the acceleration.

In a business context, rate of change is signaled by the term marginal. The derivative of a cost function is the rate of change of the cost, or marginal cost. The derivative of the profit function is the marginal profit.

Horizontal tangent lines exist where the derivative of the function equals zero. By definition, the derivative gives the slope of the tangent line. Since horizontal lines have a slope of zero, when the derivative is zero, the tangent line is horizontal.

To review, see Introduction to Derivatives.

 

3d. Estimate a tangent line slope and instantaneous rate of change from the graph of a function

  • How would one estimate the slope of a tangent line to a point on a graph?
  • Why is the slope equal to zero especially important for tangent lines?
  • What does it mean if the slope of the tangent line is equal to zero?

The slope of a tangent line to a point can be estimated from a graph by drawing a line that touches the graph at a single point and trends in the direction that the graph is going at that instant. See Figure 3d.1. You can also imagine drawing a secant line and drawing the two points closer and closer together. By extending the line, it is possible to estimate two points on the line in order to calculate an equation that approximates the tangent line.

Figure 3d.1.

Figure 3d.1.

To review, see Introduction to Derivatives.

 

3e. Write the equation of the line tangent to the graph of a function f(x)

  • What is the equation to find the slope of a tangent line at a given point?
  • What is the point-slope form of the line? The slope-intercept form?
  • How does the slope of the tangent line relate to the derivative of the function?
  • What are all the steps in the process to find the equation of the tangent line to a graph at a point?

The slope of the tangent line is the derivative of the function at the point. We can find the slope exactly at any point if we have the equation of the graph. Using the difference quotient equation, we can find the slope of the tangent line (the instantaneous rate of change), and with the slope and the ordered pair from the graph itself, we can write the equation of the tangent line.

To review, see Introduction to Derivatives.

 

3f. Calculate the derivatives of the elementary functions

  • What is the power rule?
  • What is the derivative for sin(x)? What is the derivative for cos(x)?
  • What are the circumstances that can produce a non-differentiable function?
  • What are the product and quotient rules?
  • What are some properties of derivatives?
  • What are the derivatives of other trigonometric functions?
  • What is the chain rule?
  • What is the derivative of an exponential function?
  • What is the derivative of a logarithmic function?

While the derivative of functions can be calculated from the definition, different functions produce different patterns when applying the definition. To quickly calculate derivatives, we learn the patterns, or short-cuts, rather than going through the longer process.

As functions become more complex, we can layer these rules on top of each other, from simple rules to more complex rules, that allow us to calculate the derivative of every type of differentiable function.

To review, watch Applying the Product Rule for Derivatives and Chain Rule Examples. Then, see Derivatives, Properties and Formulas, Derivative Patterns, and Some Applications of the Chain Rule.

 

3g. Calculate second and higher derivatives and state what they measure

  • What is the process for calculating a second derivative? A third derivative?
  • Describe at least two notations for a second and third derivative.
  • What is the meaning of the second derivative of a position function?

Higher derivatives exist and can have many uses as we try to understand the behavior of functions in greater detail. We can find higher derivatives by taking derivatives of derivatives. To find the second derivative, take the derivative of the first derivative. To find the third derivative, take the derivative of the second derivative, and so on. Position functions give us the clearest common meaning of a second derivative: the second derivative of position is acceleration (how fast the rate of change, velocity, is changing).

To review, see Derivative Patterns.

 

3h. Differentiate compositions of functions using the chain rule

  • What is the chain rule?
  • What are some examples of functions that require the use of the chain rule to find their derivative?
  • How do you write the chain rule in Leibniz's notation?
  • What is the chain rule in composition of functions form?

The chain rule is a method of finding the derivatives for complex functions built up from function composition. The chain rule is also useful when using tables of derivatives for special functions because it allows you to recognize a pattern and apply it to a wide variety of similar functions with the same pattern.

To review, see The Chain Rule and Some Applications of the Chain Rule.

 

3i. Use the chain rule to solve applied questions

  • What are some applications where the chain rule can be applied?

To solve applied problems, first you need to identify the kind of problem. Set up the equation to model the scenario if it’s not given in the problem itself (you may need to apply some algebra skills here). Take the derivative using the chain rule as needed and answer the question. Problems may require one derivative (velocity, rate of change, etc.) or may require more than one derivative

To review, see The Chain Rule and Some Applications of the Chain Rule.

 

3j. Calculate the derivatives of functions given as parametric equations and interpret their meanings geometrically and physically

  • What is a parametric equation?
  • How do parametric equations relate to traditional functions?
  • How do you calculate \frac{dy}{dx} for a set of parametric equations x(t) and y(t)?

Parametric equations can describe a path that is not a function using two equations, x(t) and y(t), that change with time and are independent functions. See Figure 3j.1. Because each component is a function, derivatives can be found using normal derivative rules and then recombined to find the slope of the tangent line to a curve at a specified point.

Figure 3i.1.

Figure 3j.1.

To review, see Some Applications of the Chain Rule.

 

3k. State whether a function, given by a graph or formula, is continuous or differentiable at a point or on an interval

  • What are three features of a graph that can make the function non-differentiable?
  • Draw examples of three functions that are not differentiable at x=0.
  • What are some examples of functions that are differentiable everywhere?
  • On what interval(s) is the function f(x)=1/x differentiable?

A function is considered differentiable on any interval where it is smooth, continuous, and contains no cusps or vertical tangent lines. A differentiable function has a continuous first derivative. While some common functions are not differentiable everywhere, they are differentiable on some interval, and so it is possible to apply our differentiation rules and properties as long as we avoid any non-differentiable points. See Figure 3k.1.

Figure 3j.1.

Figure 3k.1.

To review, see Derivatives, Properties, and Formulas.

 

3l. Solve related rate problems using derivatives

  • How are related rate problems related to an application of the chain rule?
  • What are the steps to solving a related rate problem?

Related rate problems are often based on geometric relationships for area or volume. The first step is to write an equation that relates the variables (not time) to each other. The next step is to take the derivative of your equation with respect to time using the chain rule.

Finally, substitute the available information into the derivative equation and solve for the missing piece. Related rate problems allow us to describe how one variable changes with respect to another variable, which is itself changing over time: such as the size of the circle as a ripple in a pond grows after dropping in a rock. See Figure 3l.1.

Figure 3k.1.

Figure 3l.1.

To review, see Related Rates.

 

3m. Approximate the solutions of equations by using derivatives and Newton's method

  • What are the steps of Newton's Method?
  • What are you trying to find when you apply Newton's Method?
  • What are some situations where Newton's Method will not work?

Newton's Method is a means of estimating the root (zero) of a function using derivatives. An initial point is guessed. The ratio of the original function divided by its derivative at the same point is subtracted from the initial guess to obtain a new guess. The process is repeated until the points remain sufficiently stable.

Newton's method can produce several errors such as when the derivative is zero. The initial guess may be a poor one, sending the point to infinity. Repeating loops or chaotic behavior can occur. It may be possible to correct these errors by guessing a different initial starting point or observing the graph of the function to make a guess closer to the correct value.

To review, see Newton's Method for Finding Roots.

 

3n. Approximate the values of difficult functions by using derivatives

  • What are some reasons why a linear approximation to a function may be sufficient?
  • What is the formula for a linear approximation?

Sometimes it is convenient to make a linear approximation to a complex function near a point. Linear functions are easy to work with and can be calculated by hand relatively easily. If you are estimating a nearby point, the difference between the estimate made from the tangent line and the true value will remain small. In these cases, we can estimate the change in the y-value as Δy as the derivative of the function at that point (the slope) times the change in the x-value as Δx, as long as Δx remains small.

To review, see Linear Approximation and Differentials.

 

3o. Calculate the differential of a function using derivatives and show what the differential represents on a graph

  • What is an example of a difficult function and a value that can be approximated by differentials?
  • What are some reasons why a linear approximation to a function may be sufficient?
  • How do the formulas for the differential differ from the formula for a linear approximation?

Sometimes it is convenient to make a linear approximation to a complex function near a point. Linear functions are easy to work with and can be calculated by hand relatively easily. If you are estimating a nearby point, the difference between the estimate made from the tangent line and the true value will remain small.

In these cases, we can estimate the change in the y-value as Δy as the derivative of the function at that point (the slope) times the change in the x-value as Δx, as long as Δx remains small. See Figure 3o.1.

Figure 3n.1.
Figure 3o.1.

To review, see Linear Approximation and Differentials.

 

3p. Calculate the derivatives of really difficult functions by using the methods of implicit differentiation and logarithmic differentiation

  • How does implicit differentiation relate to the chain rule?
  • What are the steps to performing logarithmic differentiation?

In implicit differentiation, we are unable to solve an equation explicitly, so we can apply more common differentiation rules. Instead, we assume that y is a function of x, and when we take the derivative term-by-term, we apply the chain rule any time we take the derivative of y and apply the product or quotient rule whenever x and y appear in the same term. We can then solve for \frac{dy}{dx} to find an expression for the slope of the tangent line at any point on the curve that depends on both variables (because the function is not explicit, there may be points on the curve with the same x but different y values).

Logarithmic differentiation is a technique we can use for functions where the base of an exponential function contains x, as does the exponent of the function. Since we do not have any rules for a function like xx, this process allows us to apply logarithm rules to convert the exponent into a product, and then proceed with our more common rules. In implicit differentiation, we are unable to solve an equation explicitly for y so that we can apply more common differentiation rules.

Instead, we assume that y is a function of x, and when we take the derivative term-by-term, we apply the chain rule any time we take the derivative of y, and the product or quotient rule whenever and appear in the same term. We can then solve for \frac{dy}{dx} to find an expression for the slope of the tangent line at any point on the curve that depends on both variables (because the function is not explicit, there may be points on the curve with the same x but different y values.

Logarithmic differentiation is a technique we can use for functions where the base of an exponential function contains x, as does the exponent of the function. Since we do not have any rules for a function like xx, this process allows us to apply logarithm rules to convert the exponent into a product, and then proceed with our more common rules.

To review, see Implicit and Logarithmic Differentiation.

 

Unit 3 Vocabulary

  • Acceleration
  • Chain rule
  • Composite function
  • Continuous function
  • Cusp (corner)
  • Derivative
  • Differentiable
  • Differential
  • Exponential function
  • First derivative
  • Horizontal tangent line
  • Implicit differentiation
  • Instantaneous rate of change
  • Logarithmic differentiation
  • Logarithmic functions
  • Magnification factor
  • Marginal cost
  • Marginal profit
  • Newton's Method
  • Parametric equations
  • Product rule
  • Quotient rule
  • Rate
  • Related rates
  • Second derivative
  • Slope
  • Smooth function
  • Tangent line
  • Trigonometric functions
  • Velocity
  • Vertical tangent line