Definition of the derivative

For our purposes, we use the definition of the derivative as a slope of a curve at a point or, simply, the slope of the tangent line.

This figure is of the graph of a function $y=f(x)$ showing two points $\mathrm{P}=(x, f(x))$ and $\mathrm{Q}=(x+\Delta x, f(x+\Delta x))$. The line in green is the tangent line at point $\mathrm{P}$. The line in blue is a secant line between the points $\mathrm{P}$ and Q. As $\Delta x$ tends to 0 , the following equation represents the slope of the tangent line at $\mathrm{P}$.

where, if this limit exists then we say that $f$ is differentiable at $x$ (note that $x$ is treated as a constant when taking the limit). We call the above limit a derivative and is defined formally as:

We have the following theorem stating the relationship between continuous and differentiable functions: if $f(x)$ is differentiable at a point $x=a$, then it is continuous at $x=a$. As an example, consider the Heaviside function given by:

With its graph shown for $x \in[-4,4]$ (A solid marker indicates that the endpoint $x=0$ is included while a hollow marker indicates that it is not included):

This function is a piecewise-defined function called the Heaviside function or step function often denoted by $H_{0}(x)$. The function has a jump at $x=0$ and is said to be discontinuous at $x=0$. We conclude that the function is discontinuous at $x=0$; it follows that $H^{\prime}(0)$ does not exist and $H(x)$ is not differentiable at $x=0$. We say that the function is singular at $x=0$.

Further, any differentiable function $f(x)$ must be continuous for all $x$ in its domain. Note that the converse is not true: continuity does not ensure differentiability. For instance, the function $y=|x|$ is continuous at $x=0$ (we can easily show that the two, one-sided limits as $x \rightarrow 0^{+}$and $x \rightarrow 0^{-}$are both 0 ). However, the function is not differentiable at $x=0$ (it is differentiable everywhere else in its domain). To see this, we examine the two, one-sided limits of the difference quotient

recall that in the limit $\Delta x \rightarrow 0$, the above quotient represents the derivative of the function $f$. Now if the one-sided limits disagree (i.e. as $\Delta x \rightarrow 0^{+}$and $\left.\Delta x \rightarrow 0^{-}\right)$then we conclude that the two-sided limit of the difference quotient does not exist. Indeed, if $\Delta x>0,|\Delta x| / \Delta x=1$ hence at $x=0$,

while if $\Delta x<0,|\Delta x| / \Delta x=-1$,

Since the two limits disagree, the two-sided limit does not exist and thus we find that the absolute function is not differentiable at $x=0$.

The existence of breaks (like the jump in the Heaviside function) or cusps (like in $y=|x|$ at $x=0$ ) are examples of non-differentiable behaviour. Another common form of nondifferentiability is the case where the limit of the difference quotient as $\Delta x \rightarrow 0$ is infinite (i.e. it does not exist); an example is $y=x^{1 / 3}$ at $x=0$. Finally we consider the oscillatory function:

This function is continuous for all real $x$. However, the function is not differentiable at $x=0$; again, by looking at the difference quotient

we conclude that the limit does not exist since it oscillates between -1 and 1 (i.e. it does not settle to a single, finite value).

Alternate notation

The derivative, $f^{\prime}(x)$ can also be denoted by $d f / d x$ or, since $y=f(x)$, it is often denoted by

Derivatives of $f$ are also expressed with the superscript notation

note that $f^{(1)}$ is $f^{\prime}(x)$ and the zeroth derivative, $f^{(0)}$ is simply the function $f$. The following notation

is used to show that the derivative is evaluated at the point $x=a$.

Differentiating from first principles

Before showing formulae and properties of derivatives that allow us to obtain derivatives easily, we first show a few examples of computing derivatives from first principles

The derivative of the following function using the definition of the derivative:

Consider

Again, starting from first principles

here we have used the identity

Then:

where we have used $\lim _{\theta \rightarrow 0}(\sin \theta / \theta)=1$. Hence, $f^{\prime}(x)=-\sin x$. Similarly, we can show that the derivative of $f(x)=\sin x$ is $f^{\prime}(x)=\cos x$

Differential operator

The differential operator used for taking the derivative is $d / d x$, sometimes also denoted as $D$. Applied to a function, it gives

Anything to the right of the operator is differentiated. The operator for higher, $n^{\text {th }}$ order derivatives is $d^{n} / d x^{n}$ or $D^{n}$. We can also use these definitions to define other operators like, for example, a linear second order differential operator, $L$ may be given as

with $a_{i}(x)(i=0,1,2)$ denoting coefficients which are continuous functions of the independent variable, $x$. Applying the operator to a function $y(x)$, it gives