Further Applications of Differentiation

Welcome, students 👋 In this lesson, you will see how differentiation goes beyond finding slopes of curves. It becomes a powerful tool for understanding motion, testing shapes of graphs, and solving real-world problems where change matters. By the end, you should be able to explain the main ideas behind further applications of differentiation, apply IB-style methods, and connect these ideas to the wider topic of calculus.

Lesson objectives:

• Understand key ideas and terminology in further applications of differentiation.
• Apply differentiation to graph behavior, motion, and optimisation.
• Use derivatives to solve problems in IB Mathematics: Analysis and Approaches HL.
• See how these ideas fit into the larger structure of calculus.

Think of differentiation like a microscope 🔬. It helps students zoom in on tiny changes. That is useful not only for finding gradients, but also for understanding whether a function is increasing, where it bends, how quickly a car is moving, and where a business has maximum profit.

Using the first derivative to study graph behavior

The derivative gives the rate of change of a function. If $f'(x)>0$, then $f(x)$ is increasing. If $f'(x)<0$, then $f(x)$ is decreasing. These facts are the foundation of many further applications of differentiation.

A critical point is a point where $f'(x)=0$ or where $f'(x)$ does not exist, provided the function itself is defined there. Critical points are important because they may be places where a function has a local maximum, local minimum, or a turning point.

A common IB method is to:

1. Find $f'(x)$.
2. Solve $f'(x)=0$.
3. Test the sign of $f'(x)$ on intervals.
4. Decide whether the function is increasing or decreasing.

For example, suppose $f(x)=x^3-3x^2+2$. Then

$$f'(x)=3x^2-6x=3x(x-2).$$

Set $f'(x)=0$ to get $x=0$ and $x=2$. Now check the sign of $f'(x)$:

• If $x<0$, then $f'(x)>0$.
• If $0<x<2$, then $f'(x)<0$.
• If $x>2$, then $f'(x)>0$.

So $f(x)$ increases, then decreases, then increases again. That means there is a local maximum at $x=0$ and a local minimum at $x=2$.

This sign analysis is powerful because it works even when the graph is difficult to sketch directly. In real life, it could represent a company’s revenue function, where increasing means more revenue and decreasing means less revenue.

Stationary points, maxima, and minima

A stationary point is a point where $f'(x)=0$. Stationary points can be classified using the first derivative test or the second derivative test.

The first derivative test checks whether $f'(x)$ changes sign around the stationary point. If $f'(x)$ changes from positive to negative, the function has a local maximum. If $f'(x)$ changes from negative to positive, it has a local minimum.

The second derivative test uses $f''(x)$:

• If $f'(a)=0$ and $f''(a)>0$, then $f(x)$ has a local minimum at $x=a$.
• If $f'(a)=0$ and $f''(a)<0$, then $f(x)$ has a local maximum at $x=a$.
• If $f''(a)=0$, the test is inconclusive.

For the same function $f(x)=x^3-3x^2+2$,

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

At $x=0$, $f''(0)=-6<0$, so $x=0$ is a local maximum.

At $x=2$, $f''(2)=6>0$, so $x=2$ is a local minimum.

students, this is a classic IB skill because you are not only calculating derivatives, but interpreting what they mean about the original function. That interpretation is often what earns the full marks ✨.

Concavity and points of inflection

The second derivative also tells us about concavity.

• If $f''(x)>0$, the graph is concave up.
• If $f''(x)<0$, the graph is concave down.

A point of inflection is a point where the graph changes concavity. This often happens where $f''(x)=0$ or where $f''(x)$ does not exist, but you must always check that the concavity really changes sign.

Using the earlier example,

$$f''(x)=6x-6=6(x-1).$$

So $f''(x)=0$ at $x=1$. If $x<1$, then $f''(x)<0$, so the graph is concave down. If $x>1$, then $f''(x)>0$, so the graph is concave up. Therefore, $x=1$ is a point of inflection.

Concavity can help describe shapes in contexts like bridge design, population growth, or profit curves. For example, if a graph of sales is concave up, the rate of increase is itself increasing. That means business growth is speeding up.

Optimisation: making the best choice

One of the most useful applications of differentiation is optimisation. This means finding the maximum or minimum value of a quantity.

Typical IB optimisation steps are:

1. Define variables carefully.
2. Write the quantity to be maximised or minimised as a function.
3. Use any given condition to reduce the function to one variable.
4. Differentiate and solve $f'(x)=0$.
5. Test the critical points and endpoints if needed.

Example: suppose a farmer has $100$ m of fencing and wants to enclose a rectangular area beside a river, so one side does not need fencing. Let the width be $x$ and the length be $y$. Then the fencing condition is

$$2x+y=100.$$

The area is

$$A=xy.$$

Substitute $y=100-2x$:

$$A(x)=x(100-2x)=100x-2x^2.$$

Differentiate:

$$A'(x)=100-4x.$$

Set $A'(x)=0$:

$$100-4x=0 \Rightarrow x=25.$$

Then

$$y=100-2(25)=50.$$

So the maximum area occurs when the rectangle has width $25$ m and length $50$ m. This is a great example of how calculus answers a practical question 📦.

In HL problems, the algebra can be more advanced, but the structure is usually the same. Carefully translating words into equations is often the hardest part.

Related rates and motion problems

Another important application of differentiation is kinematics, the study of motion.

If $s(t)$ is position, then:

• $v(t)=\dfrac{ds}{dt}$ is velocity.
• $a(t)=\dfrac{dv}{dt}=\dfrac{d^2s}{dt^2}$ is acceleration.

These relationships are essential in IB Calculus. For example, if

$$s(t)=t^3-6t^2+9t,$$

then

$$v(t)=3t^2-12t+9,$$

and

$$a(t)=6t-12.$$

To find when the object is at rest, solve $v(t)=0$:

$$3t^2-12t+9=0,$$

$$t^2-4t+3=0,$$

$$ (t-1)(t-3)=0.$$

So the object is at rest at $t=1$ and $t=3$.

To find when it changes direction, check the sign of $v(t)$ around those times. If velocity changes from positive to negative or vice versa, the object reverses direction. This is a real-world way calculus helps describe a moving bike, a train, or a ball thrown into the air 🚴‍♀️.

You may also see related rates problems, where two quantities change with time and are connected by a formula. A classic example is a sphere whose volume changes as its radius changes. If

$$V=\frac{4}{3}\pi r^3,$$

then differentiating with respect to $t$ gives

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

This links how fast the radius is changing to how fast the volume is changing.

Connecting local and global behavior

Further applications of differentiation are not separate from the rest of calculus. They build on the basic idea of a derivative as a rate of change and extend it into a full tool for analysing functions.

When students studies a function in IB, you often combine several ideas:

• Algebraic manipulation to simplify the function.
• Differentiation to find critical points.
• Second derivatives to study concavity.
• Graph interpretation to describe behavior.
• Context to explain the answer in a meaningful way.

This is why calculus is so connected. Differentiation tells us about instant change, and integration often tells us about total accumulation. For example, velocity from position uses differentiation, while displacement from velocity uses integration. Together, they help describe a complete motion story.

A strong IB solution usually includes both calculation and explanation. If you only write the derivative but do not interpret it, the answer may be incomplete. If you interpret without showing working, the answer may also be incomplete. Good mathematical communication matters 🧠.

Conclusion

Further applications of differentiation show how a derivative can reveal much more than slope. It can identify intervals of increase and decrease, classify maxima and minima, describe concavity, locate inflection points, solve optimisation problems, and model motion. These ideas are central to IB Mathematics: Analysis and Approaches HL because they connect symbolic calculation with meaningful interpretation. When students understands these links, calculus becomes a powerful way to analyse change in mathematics and in the real world.

Study Notes

• $f'(x)$ gives the rate of change of $f(x)$.
• If $f'(x)>0$, then $f(x)$ is increasing; if $f'(x)<0$, then $f(x)$ is decreasing.
• A critical point occurs where $f'(x)=0$ or $f'(x)$ does not exist.
• A stationary point is a point where $f'(x)=0$.
• Use the first derivative test to classify maxima and minima by sign changes in $f'(x)$.
• Use the second derivative test: if $f'(a)=0$ and $f''(a)>0$, there is a local minimum; if $f''(a)<0$, there is a local maximum.
• Concavity is determined by $f''(x)$: concave up if $f''(x)>0$, concave down if $f''(x)<0$.
• A point of inflection is where concavity changes.
• Optimisation problems usually involve defining a function, using a constraint, differentiating, and testing critical points.
• In kinematics, $v(t)=\dfrac{ds}{dt}$ and $a(t)=\dfrac{dv}{dt}$.
• Always interpret your answer in context, especially in IB exam questions.
• Differentiation is one part of the larger calculus picture, alongside integration and modelling.