The derivative of a function at a certain point is a measure of the rate at which that function is changing as an argument undergoes change. That is, a derivative embodies in terms of mathematics a rate of change. A derivative is the computation of the instantaneous slopes of f(x) at every point x. This corresponds to the slopes of the tangents to the graph of said function at said point; the slopes of such tangents can be approximated by a secant. Derivatives can also be used to compute concavity.
Functions do not have derivatives at points where they have either a vertical tangent or a discontinuity.
Differentiation and differentiability
Differentiation can be used to determine the change which something undergoes as a result of something else changing, if a mathematical relationship between two objects has been determined. The derivative of f(x) is written in several possible ways: f ′(x) (pronounced f prime of x), d/dx[f(x)] (pronounced d by d x of f of x or d d x of f of x), df/dx (pronounced d f by d x or d f d x), or Dxf (pronounced d sub x of f). The last three symbolisms are useful in considering differentiation as an operation on functions. In that context, the symbols d/dx and Dx are called differential operators.
A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at c, then there is no slope and the function is therefore not differentiable at c; however, even if a function is continuous at c, it may not be differentiable.
The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.
Newton's difference quotient
Derivatives are defined by taking the limit of the slope of secant lines as they approach a tangent line. Simply put, the derivative of a function will show the slope of the tangent line to any given point x, thus allowing it to be used to calculate other definitions of f(x).
It is hard to directly find the slope of the tangent line to a given function because we only know one point on it, the point where it is tangent to the function. Instead we will approximate the tangent line by secant lines. When we take the limit of the slopes of the nearby secant lines, we will get the slope of the tangent line.
To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is
This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:
If the derivative of f exists at every point x, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x.
Since immediately substituting 0 for ∆x results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens very easily for polynomials; see calculus with polynomials. For almost all functions, however, the result is a mess. Fortunately there are general rules which make it easy to differentiate most functions that are easy to write down; see below.
See Derivative (examples) for some examples of how to use this quotient.
The alternative difference quotient
[ed note: to relate to earlier concepts, h = ∆x. Do not be confused, however, as c != x]
Above, the derivative of f(x) (as defined by Newton) was described as the limit, as ∆x approaches zero, of [f(x + x+∆x) - f(x)] / ∆x. An alternative explanation of the derivative can be derived from Newton's quotient. Using the above; the derivative, at c, equals the limit, as ∆x approaches zero, of [f(c + ∆x) - f(c)] / ∆x; if one then lets ∆x = x - c (and c + ∆x = x); then, x approaches c (as ∆x approaches zero); thus, the derivative equals the limit, as x approaches c, of [f(x) - f(c)] / (x - c). This definition is used for a partial proof of the Chain Rule.
Notations for differentiation
- f ′(a) for the first derivative,
- f ″(a) for the second derivative,
- f ″′(a) for the third derivative and then
- f(n)(a) for the nth derivative (n > 3).
For the function whose value at each x is the derivative of f(x), we write f ′(x). Similarly, for the second derivative of f we write f ″(x), and so on.
The other common notation for differentiation is due to Leibniz. For the function whose value at x is the derivative of f at x, we write:
We can write the derivative of f at the point a in two different ways:
If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as:
Higher derivatives are expressed as
for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:
which we can loosely write as:
Dropping brackets gives the notation above.
Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "d" terms appear symbolically to cancel:
However, it is important to remember that the "d" terms cannot literally cancel, because on their own they are undefined. They are only defined when used together to express a derivative.
Newton's notation for differentiation was to place a dot over the function name:
and so on.
Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.
- For logarithmic functions:
- For trigonometric functions
- The derivative of sin x is cos x.
- The derivative of cos x is -sin x.
- The derivative of tan x is sec2 x.
- The derivative of csc x is -(csc x)(cot x).
- The derivative of sec x is (sec x)(tan x).
- The derivative of cot x is -csc2 x.
Arguably the most important application of calculus to physics is the concept of the "time derivative" — the rate of change over time — which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration.
For example, if an object's position p(t) = - 16t2 + 16t + 32; then, the object's velocity is ; the object's acceleration is ; and the object's jerk is p'''(t) = 0.
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant Rule: The derivative of any constant is zero.
- Linearity: (af + bg)' = af ' + bg' for all functions f and g and all real numbers a and b.
- General Power Rule (Polynomial rule): If f(x) = xr, for some real number r; f'(x) = rxr - 1.
- Product Rule: (fg)' = f 'g + fg' for all functions f and g.
- Quotient Rule: (f/g)' = (f 'g - fg')/(g2) unless g is zero.
- Chain Rule: If f(x) = h(g(x)), then f '(x) = h'[g(x)] * g'(x).
- Inverse functions and differentiation: If y = f(x), x = f - 1(y), and f(x) and its inverse are differentiable, then for cases in which when , dy / dx = 1 / (dx / dy)
- Derivative of one variable with respect to another when both are functions of a third variable : Let x = f(t) and y = g(t). Now Δy / Δx = (Δy / Δt) / (Δx / Δt)
- Implicit differentiation: If f(x,y) = 0 is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).
In addition, the derivatives of some common functions are useful to know. See the table of derivatives.
As an example, the derivative of
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x3 has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither.
In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3).
Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people say the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.
The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also Holomorphic function.
- table of derivatives
- covariant derivative
- Lie derivative
- exterior derivative
- exterior covariant derivative
- derivation (abstract algebra)
- functional derivative
- Kähler differential
- Pincherle derivative
- Fréchet derivative
- partial derivative
- directional derivative
- total derivative
- convective derivative
- inner derivative
- WIMS Function Calculator makes online calculation of derivatives.