The process of differentiation is used to find the rate of change.

Graphically this is the equivalent of finding the gradient of a function.

For a linear graph the gradient is constant, for example, the gradient of the graph of [latex]y=3x-7[/latex] is [latex]3[/latex].

However, the graph of a function such as [latex]y=x^2[/latex], is a curve and therefore its gradient is always changing. In this case the process of differentiation will give us a formula that can be used to find the instantaneous gradient at any particular point on the curve. This is also sometimes defined as the gradient of the tangent to the curve at a particular point.

Since differentiation measures the rate of change, it has applications in numerous fields, wherever we are measuring variation, for example distance, speed and acceleration, temperature, profit and loss, growth and decay.

Do you need this chapter?

Below is a quiz on Differentiation. If you can answer all these questions, then you can skip this chapter as you already have the requisite knowledge.

If you need to review this topic, continue reading.

Differentiation from first principles

Differentiation from first principles is the process of finding the derivative function using the definition: \begin{align*}f'(x)&=\lim_{h\rightarrow0}\frac{f(x+ h)-f(x)}{h}.\end{align*}

This formula is a version of [latex]gradient=\dfrac{rise}{run}[/latex] or [latex]\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}[/latex] as the change in [latex]x[/latex] values approaches zero (see the chapter Linear Functions: Straight Line Graphs).

Note that the derivative of a function (that is, the gradient function) can be represented as [latex]f'(x)[/latex] or as [latex]\dfrac{dy}{dx}[/latex] or as [latex]y\,'[/latex] or as [latex]\dfrac{d}{dx}(f(x))[/latex]

Differentiation by rule

For many functions, differentiation by first principles can be a long and difficult process.  Fortunately, there is a set of rules that can be used to find the derivative of most functions without having to resort to differentiation by first principles.

Mathematics students should keep these rules with them at all times!

The first and most important rule of differentiation is:

if [latex]y=x^n[/latex] then [latex]\dfrac{dy}{dx}=nx^{n-1}[/latex]

or, if written in function notation, if [latex]f(x)=x^n[/latex] then [latex]f'(x)=nx^{n-1}[/latex]

Derivatives of polynomials

Note also that the derivative of a linear function is just a number (a constant): if [latex]y = 3x[/latex]  then  [latex]\dfrac{dy}{dx}=3.[/latex]

Also, the derivative of a constant is zero: if [latex]y = 7[/latex]  then  [latex]\dfrac{dy}{dx}=0[/latex]

And, if terms are added or subtracted, you can just differentiate each term separately. For example, if [latex]\begin{align*} y &= x^3-x^2 +3x+ 7 \end{align*}[/latex] then

\begin{align*}\dfrac{dy}{dx}&=\dfrac{d}{dx}(x^{3})-\dfrac{d}{dx}(x^{2})+ \dfrac{d}{dx}(3x)+ \dfrac{d}{dx}(7).\end{align*}

Therefore

\begin{align*}\dfrac{dy}{dx}=3x^2-2x +3.\end{align*}

Finally, note that if there is a number (coefficient) in front of the "[latex]x[/latex]" term, then this number is multiplied by the power when we differentiate. For example, if [latex]y = 5x^3-3x^2[/latex] then

\begin{align*}\dfrac{dy}{dx}&=5\left(\dfrac{d}{dx}(x^{3})\right)-3\left(\dfrac{d}{dx}(x^{2})\right)\\
&=5\left(3x^{2}\right)-3\left(2x^{1}\right)\\
&=15x^2-6x.\end{align*}

Now try the questions in the exercise below.

Differentiation of functions with negative or fractional powers

The rule, if [latex]y=x^n[/latex] then [latex]\dfrac{dy}{dx}=nx^{n-1}[/latex] also applies to powers that are not positive integers, as shown below.

Note that all these examples use the rule that we used for differentiating a polynomial function. That is, if [latex]y=x^n[/latex] then [latex]\dfrac{dy}{dx}=nx^{n-1}.[/latex]  The number in the power comes to the front of the [latex]x[/latex] term and we subtract [latex]1[/latex] from the power.

Note also that you need to know the index laws to be competent at differentiation. Go to the chapter on Indices if you need to revise this topic.

Now try the questions in the exercise below.

Differentiation of trigonometric, exponential and logarithmic functions

There are, of course, many other rules that can be used to differentiate other types of functions. Some of these are shown in the table below.

Differentiation has many applications, but before you can apply it to real world situations, you need to be familiar with the rules and competent with the mathematical process.

Try the following exercise.

Besides the rules of differentiation for basic functions, there are three more rules that you need to know to differentiate more complicated functions.  These are the product rule, the quotient rule and the chain rule.

The Product rule

The product rule is used to help differentiate a function that is made up of two other functions multiplied together. For example, we would use the product rule to differentiate functions such as, [latex]y = (x^2+x)(3x-7)[/latex] or [latex]f(x)=x\sin x[/latex] or [latex]y = x^3e^x[/latex].

The product rule tells us that the derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function.

Symbolically, if [latex]u[/latex] and [latex]v[/latex] are both functions of [latex]x[/latex], and [latex]y=uv[/latex] then

\begin{align*}
\dfrac{dy}{dx}&=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\\
\end{align*}

or, using function notation, if  [latex]f(x)=g(x)×h(x)[/latex] then

\begin{align*}
f\,'(x)=g(x)×h\,'(x)+h(x)×g\,'(x).
\end{align*}

In the example above, the derivative can be easily found without resorting to the product rule. However, this is not always the case.  In fact, often the product rule is not only helpful, but necessary.

Now try this exercise.

The Quotient rule

The quotient rule is used to differentiate a function that is made up of one function divided by another function. For example, we would use the quotient rule to differentiate functions such as, [latex]y =\dfrac {x^2+x}{3x-7}[/latex] or [latex]f(x)=\dfrac{x}{\sin x}[/latex] .

If [latex]u[/latex] and [latex]v[/latex] are both functions of [latex]x[/latex] and [latex]y=\dfrac{u}{v}[/latex] then

\begin{align*}
\frac{dy}{dx}&=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}
\end{align*}

or, using function notation, if  [latex]f(x)=\dfrac{g(x)}{h(x)}[/latex] then

\begin{align*}
f'(x)=\dfrac{h(x)×g\,'(x)-g(x)×h'(x)}{[h(x)]^2}.
\end{align*}

Now try the exercises below.

The Chain rule

The chain rule, also known as the "function of a function" rule, can be used to differentiate composite functions, such as [latex]y = (x^2+5)^6 [/latex] and [latex]y=\sin^2x[/latex]  and [latex]y=\sqrt{x^2+6}[/latex].

The chain rule tells us that if [latex]y[/latex] is a function of [latex]u[/latex] where [latex]u[/latex] is a function of [latex]x[/latex] then

\begin{align*}
\frac{dy}{dx}&=\frac{dy}{du}×\frac{du}{dx}.
\end{align*}

Another form of the chain rule that is sometimes useful is the "power rule" that tells us that if [latex]y = \left[f\left(x \right) \right]^n[/latex] then

\begin{align*}
\dfrac{dy}{dx}= n\left[f\left(x \right) \right]^{n-1} \times f'\left(x \right).
\end{align*}

Now try the exercises below.

Miscellaneous derivatives

In the following exercise, you have to determine which rule to use. This is an essential skill.