Techniques of Differentiation

Introduction

Welcome, students! In this lesson, you will explore techniques of differentiation 🧠📘, which are the methods used to find derivatives of functions that are more complicated than basic power functions. Derivatives tell us how a quantity changes, and they are one of the most important ideas in calculus. They help answer questions such as: How fast is a car moving right now? Where is a graph rising or falling? What is the steepest point on a curve? 🚗📈

By the end of this lesson, you should be able to:

explain the main ideas and terminology behind techniques of differentiation,
apply standard IB-style procedures to differentiate a range of functions,
connect differentiation techniques to graphing, motion, and optimisation,
see how this topic fits into the wider study of calculus.

In IB Mathematics: Analysis and Approaches HL, techniques of differentiation are not just about calculation. They are also about choosing the right rule, simplifying expressions carefully, and interpreting what the derivative means in context.

Why Derivative Techniques Matter

The derivative is the rate at which a function changes. If $s(t)$ gives the position of an object at time $t$, then $s'(t)$ gives its velocity. If $f(x)$ is a graph, then $f'(x)$ tells us the slope of the tangent line at a point. If a business models profit by $P(x)$, then $P'(x)$ shows how profit changes when production changes.

Simple functions can often be differentiated directly using the power rule. But in real applications, functions are usually built from sums, products, quotients, and compositions. That is why students must know several techniques:

the sum and constant multiple rules,
the product rule,
the quotient rule,
the chain rule,
differentiation of trigonometric, exponential, and logarithmic functions,
implicit differentiation when the variable is not isolated.

These tools allow you to differentiate many kinds of functions that appear in modelling and problem solving.

Core Differentiation Rules

The foundation of all differentiation is the limit definition of the derivative:

$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

This formula means the derivative is the instantaneous rate of change. It is the mathematical basis for all differentiation rules, even if you usually do not calculate derivatives directly from the limit every time.

The most common rules are:

Constant rule: $\frac{d}{dx}(c)=0$
Power rule: $\frac{d}{dx}(x^n)=nx^{n-1}$
Constant multiple rule: $\frac{d}{dx}(cf(x))=cf'(x)$
Sum and difference rule: $\frac{d}{dx}(f(x)\pm g(x))=f'(x)\pm g'(x)$

Example: differentiate $f(x)=3x^4-5x^2+7$.

Using the power rule term by term:

$$f'(x)=12x^3-10x$$

This is the simplest type of differentiation, and it is the starting point for more advanced rules.

The Product and Quotient Rules

Many functions in IB are products or quotients of simpler functions. For example, $f(x)=(x^2+1)(\sin x)$ is a product, and $g(x)=\frac{x^2+1}{x-1}$ is a quotient.

Product Rule

If $u(x)$ and $v(x)$ are differentiable, then

$$\frac{d}{dx}[u(x)v(x)]=u'(x)v(x)+u(x)v'(x)$$

A common mistake is to multiply the derivatives together. That is not correct. The product rule adds two terms, not one.

Example: let $f(x)=x^2\sin x$.

Set $u(x)=x^2$ and $v(x)=\sin x$.

Then $u'(x)=2x$ and $v'(x)=\cos x$.

So,

$$f'(x)=2x\sin x+x^2\cos x$$

Quotient Rule

If $u(x)$ and $v(x)$ are differentiable and $v(x)\neq 0$, then

$$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right)=\frac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2}$$

The numerator uses subtraction, and the denominator is squared.

Example: let $g(x)=\frac{x^2+1}{x-1}$.

Here $u(x)=x^2+1$, so $u'(x)=2x$, and $v(x)=x-1$, so $v'(x)=1$.

Therefore,

$$g'(x)=\frac{(2x)(x-1)-(x^2+1)(1)}{(x-1)^2}$$

You can simplify further if needed. In exams, clear structure is often more important than rushing to the final simplified form.

The Chain Rule and Composition

The chain rule is one of the most important techniques in differentiation 🔗. It is used when one function is inside another function. For example, $y=(3x+2)^5$ or $y=\sin(x^2)$.

If $y=f(g(x))$, then

$$\frac{dy}{dx}=f'(g(x))\cdot g'(x)$$

A helpful way to think about this is: differentiate the outside function first, keep the inside the same, then multiply by the derivative of the inside.

Example: differentiate $y=(3x+2)^5$.

The outside function is $u^5$, and the inside is $u=3x+2$.

So,

$$\frac{dy}{dx}=5(3x+2)^4\cdot 3=15(3x+2)^4$$

Example: differentiate $y=\sin(x^2)$.

The outside function is $\sin u$, and the inside is $u=x^2$.

So,

$$\frac{dy}{dx}=\cos(x^2)\cdot 2x=2x\cos(x^2)$$

The chain rule also appears in many harder problems, including implicit differentiation and applications in optimisation.

Differentiating Exponential, Logarithmic, and Trigonometric Functions

In IB Mathematics: Analysis and Approaches HL, you must know the standard derivatives of common functions.

For exponentials and logarithms:

$$\frac{d}{dx}(e^x)=e^x$$

$$\frac{d}{dx}(a^x)=a^x\ln a$$

$$\frac{d}{dx}(\ln x)=\frac{1}{x}, \quad x>0$$

For trig functions:

$$\frac{d}{dx}(\sin x)=\cos x$$

$$\frac{d}{dx}(\cos x)=-\sin x$$

$$\frac{d}{dx}(\tan x)=\sec^2 x$$

These derivatives are often combined with the chain rule.

Example: differentiate $y=e^{4x^2}$.

The outside function is $e^u$, and the inside is $u=4x^2$.

Thus,

$$\frac{dy}{dx}=e^{4x^2}\cdot 8x=8xe^{4x^2}$$

Example: differentiate $y=\ln(5x-1)$.

Using the chain rule,

$$\frac{dy}{dx}=\frac{1}{5x-1}\cdot 5=\frac{5}{5x-1}$$

These examples show why students should always identify the structure of the function before differentiating.

Implicit Differentiation and Related Skills

Sometimes a relationship between $x$ and $y$ is given without making $y$ the subject. For example:

$$x^2+y^2=25$$

This is the equation of a circle, and it is not easy to solve for $y$ in a way that gives one formula for the whole curve. In such cases, implicit differentiation is useful.

Differentiate both sides with respect to $x$:

$$\frac{d}{dx}(x^2)+\frac{d}{dx}(y^2)=\frac{d}{dx}(25)$$

This gives

$$2x+2y\frac{dy}{dx}=0$$

Now solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx}=-\frac{x}{y}$$

The key idea is that when differentiating $y$, you must multiply by $\frac{dy}{dx}$ because $y$ depends on $x$.

Implicit differentiation is useful for finding tangent lines, gradients, and points where curves have horizontal or vertical tangents. It also connects to the study of curves in the coordinate plane.

How Techniques of Differentiation Fit into Calculus

Techniques of differentiation connect closely to the rest of calculus. Once students can differentiate well, the derivative becomes a tool for many other topics:

Graph analysis: Find intervals where a function is increasing or decreasing using $f'(x)$.
Stationary points: Solve $f'(x)=0$ to locate turning points.
Optimisation: Use derivatives to find maximum or minimum values in real-life problems.
Kinematics: If $s(t)$ is position, then $v(t)=\frac{ds}{dt}$ and $a(t)=\frac{dv}{dt}$.
Further modelling: Derivatives are used in differential equations and approximation methods.

Example: if a company’s profit is $P(x)$, then solving $P'(x)=0$ may identify the number of products that gives the highest profit. In physics, if $s(t)=t^3-6t^2+9t$, then

$$v(t)=\frac{ds}{dt}=3t^2-12t+9$$

and

$$a(t)=\frac{dv}{dt}=6t-12$$

This shows how differentiation helps turn a position formula into motion information.

Conclusion

Techniques of differentiation give students a powerful toolkit for working with a wide range of functions. The core rules, such as the power rule, product rule, quotient rule, chain rule, and implicit differentiation, make it possible to find derivatives efficiently and accurately. These techniques are essential in IB Mathematics: Analysis and Approaches HL because they support graph sketching, modelling, optimisation, and kinematics. In other words, differentiation is not only about getting an answer; it is about understanding change 📊✨

Study Notes

The derivative is defined by $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$.
Use the power rule: $\frac{d}{dx}(x^n)=nx^{n-1}$.
Use the product rule: $\frac{d}{dx}[uv]=u'v+uv'$.
Use the quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}$.
Use the chain rule for compositions: $\frac{dy}{dx}=f'(g(x))g'(x)$.
Know the standard derivatives of $e^x$, $\ln x$, $\sin x$, $\cos x$, and $\tan x$.
In implicit differentiation, differentiate both sides with respect to $x$ and include $\frac{dy}{dx}$ when differentiating $y$.
Derivatives are used for tangents, motion, increasing/decreasing intervals, and optimisation.
Techniques of differentiation are a major bridge between algebraic manipulation and real-world calculus applications.