4 Applications of Derivatives

A clear progression through the main applications of derivatives: modeling changing quantities, approximating values, analyzing extrema and graph shape, applying the Mean Value Theorem, and solving optimization problems.

The derivative as an application tool

Derivatives measure instantaneous rates of change and slopes of tangent lines. Their applications follow a common pattern:

Translate the situation into variables, functions, and equations.
Differentiate to obtain rates, slopes, or curvature information.
Use derivative information to approximate values, analyze behavior, or locate candidate extrema.
Interpret the result in its original context, including units, signs, domain restrictions, and endpoints.

The first derivative describes instantaneous change and tangent-line slope. The second derivative describes how the first derivative changes, revealing concavity and helping classify some extrema.

Takeaway: Derivative applications are most reliable when symbolic modeling, differentiation, testing, and contextual interpretation are treated as connected steps.

Relating changing quantities

When two or more quantities change with time and are connected by an equation, the situation is a . The equation should be differentiated with respect to time before numerical values are substituted.

A dependable workflow

Assign variables to every changing quantity.
Record the known rates and identify the rate to find.
Write an equation relating the variables.
Differentiate with respect to time, using the chain rule.
Substitute known numerical values only after differentiating.
Solve for the requested rate and state appropriate units.

For a spherical balloon, the volume-radius relationship is

V=\frac{4}{3}\pi r^3.

Differentiating with respect to time gives

\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}.

If $\frac{dV}{dt}=2$ cubic centimeters per second and $r=3$ centimeters, then

2=4\pi(3)^2\frac{dr}{dt}.

Therefore,

\frac{dr}{dt}=\frac{1}{18\pi}\text{ cm/s}.

The positive sign indicates that the radius is increasing. Useful geometric relationships include

A=\pi r^2,\qquad C=2\pi r,\qquad V=\frac{4}{3}\pi r^3,

V=\pi r^2h,\qquad a^2+b^2=c^2.

Takeaway: Keep changing quantities symbolic through differentiation; substitute values afterward so their rates do not disappear.

Approximating nearby values

A tangent line provides a local approximation to a differentiable function. at $x=a$ is

L(x)=f(a)+f'(a)(x-a),

so for an input close to $a$ ,

f(x)\approx L(x).

To approximate $\sqrt{4.1}$ , choose $f(x)=\sqrt{x}$ and $a=4$ . Then

f(4)=2,\qquad f'(4)=\frac{1}{4}.

Thus,

L(x)=2+\frac{1}{4}(x-4),

and

\sqrt{4.1}\approx L(4.1)=2+\frac{1}{4}(0.1)=2.025.

Differentials express the same local-change idea. If $y=f(x)$ , then

dy=f'(x)\,dx.

For a small input change, the actual output change satisfies $\Delta y\approx dy$ . For the area of a circle,

A=\pi r^2\quad\Longrightarrow\quad dA=2\pi r\,dr.

If $dy$ estimates an error in $y$ , then the approximate relative and percentage errors are

\frac{|dy|}{|y|}\qquad\text{and}\qquad 100\frac{|dy|}{|y|}\%.

Takeaway: Use a tangent-line model near a convenient base point, and use differentials to estimate small changes or propagated measurement errors.

Finding and classifying extrema

A occurs at a domain value $c$ where

f'(c)=0

or where $f'(c)$ does not exist. Critical points are candidates for local maxima and minima, but they must be tested.

The First Derivative Test examines the sign of the derivative around a :

A change from positive to negative indicates a local maximum.
A change from negative to positive indicates a local minimum.
No sign change indicates neither type of local extremum.

If $f'(c)=0$ and the second derivative exists nearby, the Second Derivative Test gives a quicker classification:

If $f''(c)>0$ , the function has a local minimum.
If $f''(c)<0$ , the function has a local maximum.
If $f''(c)=0$ , the test is inconclusive.

For absolute extrema on a closed interval, continuity guarantees that both an absolute maximum and an absolute minimum exist. Evaluate the function at every in the interior and at both endpoints, then compare all values.

Takeaway: Finding candidates is only the first step; classification and comparison determine the actual extrema.

Connecting average and instantaneous change

The connects average change over an interval with instantaneous change at an interior point. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then some $c\in(a,b)$ satisfies

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

The right-hand side is the slope of the secant line through the endpoints. For $f(x)=x^2$ on $[1,3]$ , the average rate is

\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4.

Because $f'(x)=2x$ , the matching interior point solves

2c=4,

so $c=2$ .

Important consequences include:

If the derivative is zero throughout an interval, the function is constant there.
If the derivative is positive throughout an interval, the function is increasing there.
If the derivative is negative throughout an interval, the function is decreasing there.
If two functions have the same derivative on an interval, they differ by a constant on that interval.

Takeaway: Under the theorem's hypotheses, a derivative sign controls monotonic behavior across an entire interval, not merely at one point.

Reading graph shape from derivative signs

The sign of the first derivative determines whether a function rises or falls. To analyze increasing and decreasing behavior:

Determine the domain.
Compute the first derivative.
Find values where the derivative is zero or undefined.
Use those values to divide the domain into intervals.
Test the derivative sign on each interval.

For

f(x)=x^3-3x^2+1,

we have

f'(x)=3x(x-2).

The critical numbers are $x=0$ and $x=2$ . The derivative is positive on $(-\infty,0)$ , negative on $(0,2)$ , and positive on $(2,\infty)$ . Thus, the function increases, then decreases, then increases again. The sign changes identify a local maximum at $x=0$ and a local minimum at $x=2$ .

The second derivative describes changes in slope. If $f''(x)>0$ , the graph is concave up; if $f''(x)<0$ , it is concave down. An requires an actual change in concavity, not merely a value where the second derivative is zero.

For the same function,

f''(x)=6x-6.

The second derivative changes from negative to positive at $x=1$ , so the is

(1,f(1))=(1,-1).

Takeaway: Use a sign chart for the first derivative to determine monotonicity and a sign chart for the second derivative to determine concavity.

Building a derivative-based sketch

A derivative-based curve sketch combines domain information, function values, and derivative tests. A useful sequence is:

Determine the domain and identify discontinuities.
Find intercepts and other useful points.
Examine end behavior and asymptotes when relevant.
Find critical points from the first derivative.
Use a first-derivative sign chart for increasing and decreasing intervals.
Find candidates for inflection points from the second derivative.
Use a second-derivative sign chart for concavity.
Classify extrema and verify changes in concavity.
Combine the results into a graph that respects the original domain.

For a rational function, values that make the denominator zero are excluded from the domain and may produce vertical asymptotes. Derivative conclusions must therefore be checked against domain restrictions.

Takeaway: A reliable sketch comes from combining several independent checks rather than relying on a few plotted points.

Turning constraints into optimal decisions

begins by translating a practical question into an objective function and constraints. The general strategy is:

Describe the situation with a diagram or variables.
Identify the quantity to maximize or minimize.
Write the objective function.
Use the constraints to reduce it to one variable.
Determine the feasible domain.
Find critical points and evaluate endpoints when appropriate.
Compare candidate values and interpret the result with units.

For a rectangle with perimeter $40$ meters, let the length be $x$ and width be $y$ . The constraint is

2x+2y=40,\qquad y=20-x.

The area becomes

A(x)=x(20-x)=20x-x^2.

Differentiating gives

A'(x)=20-2x.

The satisfies

20-2x=0,\qquad x=10.

Then $y=10$ , and because

A''(x)=-2<0,

this point gives a maximum. The largest area is

A=100\text{ m}^2.

Other objective functions include revenue, profit, distance, surface area, volume, and travel time. For example,

R(x)=x\,p(x),\qquad P(x)=R(x)-C(x),\qquad T=\frac{\text{distance}}{\text{speed}}.

In economic applications, marginal cost, marginal revenue, and marginal profit are represented by $C'(x)$ , $R'(x)$ , and $P'(x)$ . A profit maximum may occur where marginal revenue equals marginal cost, provided the domain and maximum tests support that conclusion.

Takeaway: A derivative supplies candidates, but constraints, endpoints, comparison, and contextual interpretation establish the final answer.

Motion, marginal change, and numerical roots

Derivatives also describe motion and provide numerical tools.

For position $s(t)$ , velocity and acceleration are

v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).

An object moves forward when velocity is positive and backward when velocity is negative. It speeds up when velocity and acceleration have the same sign and slows down when their signs differ.

In economics, derivatives describe marginal quantities:

C'(x)=\text{marginal cost},\qquad R'(x)=\text{marginal revenue},\qquad P'(x)=\text{marginal profit}.

uses tangent-line intersections to approximate roots of $f(x)=0$ :

x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.

It is most effective when the starting approximation is close to a root and the derivative does not vanish nearby.

These applications fit together as one strategy: model the situation, differentiate, analyze signs or approximations, test candidates, and interpret the result in context. Related rates track changing quantities; estimates nearby values; derivative tests reveal extrema and graph shape; and uses these ideas to make practical decisions.

Takeaway: The same derivative information can describe motion, marginal change, root approximations, graph shape, and optimal choices.