Application of Derivatives

Applications of Derivatives in Calculus

Being able to solve the related rates problem discussed above is just one of many applications of derivatives you learn in calculus. You will also learn how derivatives are used to:

• find tangent and normal lines to a curve, and

• find maximum and minimum values.

You will then be able to use these techniques to solve optimization problems, like maximizing an area or maximizing revenue.

Additionally, you will learn how derivatives can be applied to:

• solve complicated limits,

• make approximations, and

• to give accurate graphs.

Application of Derivatives: Tangent and Normal Lines

Derivatives are very useful tools for finding the equations of tangent lines and normal lines to a curve.

Fig. 1. Using the derivative to find the tangent and normal lines to a curve.

Tangent Lines to a Curve

The tangent line to a curve is one that touches the curve at only one point and whose slope is the derivative of the curve at that point.

For more information and examples about this subject, see our article on Tangent Lines.

Normal Lines to a Curve

The normal line to a curve is perpendicular to the tangent line. You use the tangent line to the curve to find the normal line to the curve. The slope of the normal line is:

\[ n = - \frac{1}{m} = - \frac{1}{f'(x)}. \]

Application of Derivatives: Related Rates

In many real-world scenarios, related quantities change with respect to time. If you think about the rocket launch again, you can say that the rate of change of the rocket's height, \( h \), is related to the rate of change of your camera's angle with the ground, \( \theta \). In this case, you say that \( \frac{dg}{dt} \) and \( \frac{d\theta}{dt} \) are related rates because \( h \) is related to \( \theta \).

In related rates problems, you study related quantities that are changing with respect to time and learn how to calculate one rate of change if you are given another rate of change.

There are lots of different articles about related rates, including Rates of Change, Motion Along a Line, Population Change, and Changes in Cost and Revenue.

Application of Derivatives: Linear Approximations and Differentials

When it comes to functions, linear functions are one of the easier ones with which to work. Therefore, they provide you a useful tool for approximating the values of other functions. You will build on this application of derivatives later as well, when you learn how to approximate functions using higher-degree polynomials while studying sequences and series, specifically when you study power series.

The key concepts and equations of linear approximations and differentials are:

• A differentiable function, \( y = f(x) \), can be approximated at a point, \( a \), by the linear approximation function:

  \[ L(x) = f(a) + f'(a)(x-a). \]

• Given a function, \( y = f(x) \), if, instead of replacing \( x \) with \( a \), you replace \( x \) with \( a + dx \), then the differential:

  \[ dy = f'(x)dx \]

  is an approximation for the change in \( y \).

  • The actual change in \( y \), however, is:

    \[ \Delta y = f(a+dx) - f(a). \]

• A measurement error of \( dx \) can lead to an error in the quantity of \( f(x) \). This is known as propagated error, which is estimated by:

  \[ dy \approx f'(x)dx \]

• To estimate the relative error of a quantity ( \( q \) ) you use:\[ \frac{ \Delta q}{q}. \]

For more information on this topic, see our article on the Amount of Change Formula.

Application of Derivatives: Maxima and Minima

One of the most common applications of derivatives is finding the extreme values, or maxima and minima, of a function. Once you learn the methods of finding extreme values (also known collectively as extrema), you can apply these methods to later applications of derivatives, like creating accurate graphs and solving optimization problems.

The key terms and concepts of maxima and minima are:

• Terms

  • Absolute extremum

    If a function, \( f \), has an absolute max or absolute min at point \( c \), then you say that the function \( f \) has an absolute extremum at \( c \).

  • Absolute max / absolute maximum

    If \( f(c) \geq f(x) \) for all \( x \) in the domain of \( f \), then you say that \( f \) has an absolute maximum at \( c \).

  • Absolute min / absolute minimum

    If \( f(c) \leq f(x) \) for all \( x \) in the domain of \( f \), then you say that \( f \) has an absolute minimum at \( c \).

  • Local extremum

    If a function, \( f \), has a local max or min at point \( c \), then you say that \( f \) has a local extremum at \( c \).

  • Local max / local maximum

    If there exists an interval, \( I \), such that \( f(c) \geq f(x) \) for all \( x \) in \( I \), you say that \( f \) has a local max at \( c \).

  • Local min / local minimum

    If there exists an interval, \( I \), such that \( f(c) \leq f(x) \) for all \( x \) in \( I \), you say that \( f \) has a local min at \( c \).

  • Critical point

    Based on the definitions above, the point \( (c, f(c)) \) is a critical point of the function \( f \).

  • Critical number

    If \( f'(c) = 0 \) or \( f'(c) \) is undefined, you say that \( c \) is a critical number of the function \( f \).

  • Extreme value theorem

    If the function \( f \) is continuous over a finite, closed interval, then \( f \) has an absolute max and an absolute min.

  • Fermat's theorem

    If a function \( f \) has a local extremum at point \( c \), then \( c \) is a critical point of \( f \).

• Concepts

  • It is possible for a function to have:

    • both an absolute max and an absolute min,

    • just one absolute extremum, or

    • have no absolute extremum.

  • If a function has a local extremum, the point where it occurs must be a critical point.

    • However, a function does not necessarily have a local extremum at a critical point.

  • A continuous function over a closed and bounded interval has an absolute max and an absolute min.

    • Each extremum occurs at either a critical point or an endpoint of the function.

For more information on maxima and minima see Maxima and Minima Problems and Absolute Maxima and Minima.

Application of Derivatives: The Mean Value Theorem

One of the most important theorems in calculus, and an application of derivatives, is the Mean Value Theorem (sometimes abbreviated as MVT). Like the previous application, the MVT is something you will use and build on later.

The key concepts of the mean value theorem are:

• The definition of the MVT

  If a function, \( f \), is continuous over the closed interval \( [a, b] \) and differentiable over the open interval \( (a, b) \), then there exists a point \( c \) in the open interval \( (a, b) \) such that

  \[ f'(c) = \frac{f(b)-f(a)}{b-a}. \]

• The special case of the MVT known as Rolle's theorem

  If a function, \( f \), is continuous over the closed interval \( [a, b] \), differentiable over the open interval \( (a, b) \), and if \( f(a) = f(b) \), then there exists a point \( c \) in the open interval \( (a, b) \) such that

  \[ f'(c) = 0 \]

• The corollaries of the mean value theorem

  1. Functions with a derivative of zero

     Let \( f \) be differentiable on an interval \( I \). If \( f'(x) = 0 \) for all \( x \) in \( I \), then \( f'(x) = \) constant for all \( x \) in \( I \).

  2. Constant difference theorem

     If the functions \( f \) and \( g \) are differentiable over an interval \( I \), and \( f'(x) = g'(x) \) for all \( x \) in \( I \), then \( f(x) = g(x) + C \) for some constant \( C \).

  3. Increasing and decreasing functions

     Let \( f \) be continuous over the closed interval \( [a, b] \) and differentiable over the open interval \( (a, b) \).

     1. If \( f'(x) > 0 \) for all \( x \) in \( (a, b) \), then \( f \) is an increasing function over \( [a, b] \).

     2. If \( f'(x) < 0 \) for all \( x \) in \( (a, b) \), then \( f \) is a decreasing function over \( [a, b] \).

Application of Derivatives: Derivatives and the Shape of a Graph

Building on the applications of derivatives to find maxima and minima and the mean value theorem, you can now determine whether a critical point of a function corresponds to a local extreme value. But what about the shape of the function's graph? Well, this application teaches you how to use the first and second derivatives of a function to determine the shape of its graph.

Key concepts of derivatives and the shape of a graph are:

• the first derivative test

  Say a function, \( f \), is continuous over an interval \( I \) and contains a critical point, \( c \). If \( f \) is differentiable over \( I \), except possibly at \( c \), then \( f(c) \) satisfies one of the following:

  1. If \( f' \) changes sign from positive when \( x < c \) to negative when \( x > c \), then \( f(c) \) is a local max of \( f \).

  2. If \( f' \) changes sign from negative when \( x < c \) to positive when \( x > c \), then \( f(c) \) is a local min of \( f \).

  3. If \( f' \) has the same sign for \( x < c \) and \( x > c \), then \( f(c) \) is neither a local max or a local min of \( f \).

• the Candidates Test

  This is a method for finding the absolute maximum and the absolute minimum of a continuous function that is defined over a closed interval. It consists of the following:

  1. Find all the relative extrema of the function.

  2. Evaluate the function at the extreme values of its domain.

  3. Order the results of steps 1 and 2 from least to greatest.

     • The least value is the global minimum.

     • The greatest value is the global maximum.

• test for concavity

  If \( f \) is a function that is twice differentiable over an interval \( I \), then:

  1. If \( f''(x) > 0 \) for all \( x \) in \( I \), then \( f \) is concave up over \( I \).

  2. If \( f''(x) < 0 \) for all \( x \) in \( I \), then \( f \) is concave down over \( I \).

• the second derivative test

  Suppose \( f'(c) = 0 \), \( f'' \) is continuous over an interval that contains \( c \).

  1. If \( f''(c) > 0 \), then \( f \) has a local min at \( c \).

  2. If \( f''(c) < 0 \), then \( f \) has a local max at \( c \).

  3. If \( f''(c) = 0 \), then the test is inconclusive.

     • You must evaluate \( f'(x) \) at a test point \( x \) to the left of \( c \) and a test point \( x \) to the right of \( c \) to determine if \( f \) has a local extremum at \( c \).

Application of Derivatives: Limits at Infinity and Asymptotes

Once you understand derivatives and the shape of a graph, you can build on that knowledge to graph a function that is defined on an unbounded domain. To accomplish this, you need to know the behavior of the function as \( x \to \pm \infty \). This application of derivatives defines limits at infinity and explains how infinite limits affect the graph of a function.

The key terms and concepts of limits at infinity and asymptotes are:

• Terms

  • end behavior

    The behavior of the function, \( f(x) \), as \( x\to \pm \infty \).

  • horizontal asymptote

    If \( \lim_{x \to \pm \infty} f(x) = L \), then \( y = L \) is a horizontal asymptote of the function \( f(x) \).

  • infinite limit at infinity

    The function \( f(x) \) becomes larger and larger as \( x \) also becomes larger and larger.

  • limit at infinity

    The limiting value, if it exists, of a function \( f(x) \) as \( x \to \pm \infty \).

  • oblique asymptote

    The line \( y = mx + b \), if \( f(x) \) approaches it, as \( x \to \pm \infty \) is an oblique asymptote of the function \( f(x) \).

• Concepts

  • The limit of the function \( f(x) \) is \( L \) as \( x \to \pm \infty \) if the values of \( f(x) \) get closer and closer to \( L \) as \( x \) becomes larger and larger.

  • The limit of the function \( f(x) \) is \( \infty \) as \( x \to \infty \) if \( f(x) \) becomes larger and larger as \( x \) also becomes larger and larger.

  • The limit of the function \( f(x) \) is \( - \infty \) as \( x \to \infty \) if \( f(x) < 0 \) and \( \left| f(x) \right| \) becomes larger and larger as \( x \) also becomes larger and larger.

  • For the polynomial function \( P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{1}x + a_{0} \), where \( a_{n} \neq 0 \), the end behavior is determined by the leading term: \( a_{n}x^{n} \).

    • If \( n \neq 0 \), then \( P(x) \) approaches \( \pm \infty \) at each end of the function.

  • For the rational function \( f(x) = \frac{p(x)}{q(x)} \), the end behavior is determined by the relationship between the degree of \( p(x) \) and the degree of \( q(x) \).

    • If the degree of \( p(x) \) is less than the degree of \( q(x) \), then the line \( y = 0 \) is a horizontal asymptote for the rational function.

    • If the degree of \( p(x) \) is equal to the degree of \( q(x) \), then the line \( y = \frac{a_{n}}{b_{n}} \), where \( a_{n} \) is the leading coefficient of \( p(x) \) and \( b_{n} \) is the leading coefficient of \( q(x) \), is a horizontal asymptote for the rational function.

    • If the degree of \( p(x) \) is greater than the degree of \( q(x) \), then the function \( f(x) \) approaches either \( \infty \) or \( - \infty \) at each end.

Application of Derivatives: Optimization Problems

Continuing to build on the applications of derivatives you have learned so far, optimization problems are one of the most common applications in calculus. These are defined as calculus problems where you want to solve for a maximum or minimum value of a function.

Application of Derivatives: L’Hôpital’s Rule

A powerful tool for evaluating limits, L’Hôpital’s Rule is yet another application of derivatives in calculus. This application uses derivatives to calculate limits that would otherwise be impossible to find. These limits are in what is called indeterminate forms.

The key terms and concepts of L’Hôpital’s Rule are:

• Terms

  • indeterminate forms

    When evaluating a limit, the forms \[ \frac{0}{0}, \ \frac{\infty}{\infty}, \ 0 \cdot \infty, \ \infty - \infty, \ 0^{0}, \ \infty^{0}, \ \mbox{ and } 1^{\infty} \] are all considered indeterminate forms because you need to further analyze (i.e., by using L’Hôpital’s rule) whether the limit exists and, if so, what the value of the limit is.

  • L’Hôpital’s Rule

    If two functions, \( f(x) \) and \( g(x) \), are differentiable functions over an interval \( a \), except possibly at \( a \), and \[ \lim_{x \to a} f(x) = 0 = \lim_{x \to a} g(x) \] or \[ \lim_{x \to a} f(x) \mbox{ and } \lim_{x \to a} g(x) \mbox{ are infinite, } \] then \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}, \] assuming the limit involving \( f'(x) \) and \( g'(x) \) either exists or is \( \pm \infty \).

• Concepts

  • You can use L’Hôpital’s rule to evaluate the limit of a quotient when it is in either of the indeterminate forms \( \frac{0}{0}, \ \frac{\infty}{\infty} \).

  • You can also use L’Hôpital’s rule on the other indeterminate forms if you can rewrite them in terms of a limit involving a quotient when it is in either of the indeterminate forms \( \frac{0}{0}, \ \frac{\infty}{\infty} \).

Application of Derivatives: Newton’s Method

In many applications of math, you need to find the zeros of functions. Unfortunately, it is usually very difficult – if not impossible – to explicitly calculate the zeros of these functions. Newton's method saves the day in these situations because it is a technique that is efficient at approximating the zeros of functions.

The key terms and concepts of Newton's method are:

• Terms

  • iterative process

    A process in which a list of numbers like \[ x_{0}, x_{1}, x_{2}, \ldots \] is generated by beginning with a number \( x_{0} \) and then defining \[ x_{n} = F \left( x_{n-1} \right) \] for \( n \neq 1 \).

  • Newton's method

    A method for approximating the roots of \( f(x) = 0 \). It uses an initial guess of \( x_{0} \). Each subsequent approximation is defined by the equation \[ x_{n} = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}. \]

• Concepts

  • Newton's method approximates the roots of \( f(x) = 0 \) by starting with an initial approximation of \( x_{0} \). From there, it uses tangent lines to the graph of \( f(x) \) to create a sequence of approximations \( x_1, x_2, x_3, \ldots \).

  • Failures of Newton's method:

    • This method fails when the list of numbers \( x_1, x_2, x_3, \ldots \) does not approach a finite value, or

    • when it approaches a value other than the root you are looking for.

  • Any process in which a list of numbers \( x_1, x_2, x_3, \ldots \) is generated by defining an initial number \( x_{0} \) and defining the subsequent numbers by the equation \[ x_{n} = F \left( x_{n-1} \right) \] for \( n \neq 1 \) is an iterative process.

    • Newton's method is an example of an iterative process, where the function \[ F(x) = x - \left[ \frac{f(x)}{f'(x)} \right] \] for a given function of \( f(x) \).

Application of Derivatives: Antiderivatives

Having gone through all the applications of derivatives above, now you might be wondering: what about turning the derivative process around? What if I have a function \( f(x) \) and I need to find a function whose derivative is \( f(x) \)? What application does this have?

To answer these questions, you must first define antiderivatives.

• An antiderivative of a function \( f \) is a function whose derivative is \( f \).

One of many examples where you would be interested in an antiderivative of a function is the study of motion.

The key terms and concepts of antiderivatives are:

• Terms

  • antiderivative

    A function \( F(x) \) such that \( F'(x) = f(x) \) for all \( x \) in the domain of \( f \) is an antiderivative of \( f \).

  • indefinite integral

    The most general antiderivative of a function \( f(x) \) is the indefinite integral of \( f \). The notation \[ \int f(x) dx \] denotes the indefinite integral of \( f(x) \).

  • initial value problem

    A problem that requires you to find a function \( y \) that satisfies the differential equation \[ \frac{dy}{dx} = f(x) \] together with the initial condition of \[ y(x_{0}) = y_{0}. \]

• Concepts

  • If the function \( F \) is an antiderivative of another function \( f \), then every antiderivative of \( f \) is of the form \[ F(x) + C \] for some constant \( C \).

  • Solving the initial value problem \[ \frac{dy}{dx} = f(x), \mbox{ with the initial condition } y(x_{0}) = y_{0} \] requires you to:

    • first find the set of antiderivatives of \( f \) and then

    • look for the particular antiderivative that also satisfies the initial condition.

Applications of Derivatives in Engineering

The applications of derivatives in engineering is really quite vast. To touch on the subject, you must first understand that there are many kinds of engineering. To name a few;

• Mechanical Engineering

• Civil Engineering

• Industrial Engineering

• Electrical Engineering

• Aerospace Engineering

• Chemical Engineering

• Computer Engineering

• \( \vdots \)

All of these engineering fields use calculus. They all use applications of derivatives in their own way, to solve their problems.

• An example that is common among several engineering disciplines is the use of derivatives to study the forces acting on an object. For instance,

  • Mechanical Engineers could study the forces that on a machine (or even within the machine).

  • Civil Engineers could study the forces that act on a bridge.

  • Industrial Engineers could study the forces that act on a plant.

  • Aerospace Engineers could study the forces that act on a rocket.

  • And so on.

In every case, to study the forces that act on different objects, or in different situations, the engineer needs to use applications of derivatives (and much more).

Applications of Derivatives in Economics

Even the financial sector needs to use calculus! Applications of derivatives are used in economics to determine and optimize:

• supply and demand,

• profit and cost, and

• revenue and loss.

Applications of Derivatives: Examples