Please help me explain what a derivative is if you were explaining it to a first time learner?

Question

the explaination could be used to explain to another college student possibly even one who is not in your class.

Answer 1

It is the slope of a curve for a function. It is the rate that shows how much one variable changes for each small change in the other variable.

Answer 2

Explaining the concept of a limit is a bit tricky. But I always used the following example:
Imagine you're walking down a path in a garden, and the path is marked with small, round 'paver' stones that are just touching. Now, if you come to a point where a paver stone is missing, even 'tho it isn't there you --still-- know where it would be if it --was-- there.
And that's the idea behind a limit. It's the value that a function would assume for some value of the independent variable, if the function didn't vanish (become undefined) at that point.
For example; (x² - 1)/(x - 1) is well defined except at x = 1 (where the denominator becomes 0. But how about if you divide it through to get (x² - 1)/(x - 1) = x + 1. All well and good except that you can **NOT** divide (x-1)/(x-1) when x = 1. You get 0/0 and that's a huge no-no. So the function y = (x² - 1)/(x - 1) looks just like y = x+1 **except** at x = 1 where it has a little 'hole' in it where the point (1,2) should be, but isn't.

HTH

Doug

Edit:
Hey guys......... Back to the basics. A derivative can be (and frequently is) interpreted as the 'slope' of a tangent line to a function at some point, but it's --defined-- as a limit.
f'(x) = lim δ -> 0 of [f(x + δ) - f(x)] / δ
So........ No Desert until you finish your Veggies ☺

Doug

Answer 3

The derivative is an equation derived from another equation to show the equation of a tangent line for any point on the equation. I know it's confusing, but you'll have to explain tangent lines.

Answer 4

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization.

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.

Differentiation is a method to compute the rate at which a quantity, y, changes with respect to the change in another quantity, x, upon which it is dependent. This rate of change is called the derivative of y with respect to x. In more precise language, the dependency of y on x means that y is a function of x. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = f(x), where f denotes the function.

The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = f(x) = m x + c, for real numbers m and c, and the slope m is given where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". This formula is true because;

y + Δy = f(x+ Δx) = m (x + Δx) + c = m x + c + m Δx = y + mΔx.
It follows that Δy = m Δx.

This gives an exact value for the slope of a straight line. If the function f is not a straight line, however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.



The most common approach to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis


Definition via difference quotients
Let y=f(x) be a function of x. In classical geometry, the tangent line at a real number a was the unique line through the point (a, f(a)) which did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a. The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero will give a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations.

This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line. Formally, the derivative of the function f at a is the limit of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then f is differentiable at a. Here f '(a) is one of several common notations for the derivative
More generally, a similar computation shows that the derivative of the squaring function at x = a is f '(a) = 2a.

This function does not have a derivative at the marked point, as the function is not continuous there.If y = f(x) is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f be the step function which returns a value, say 1, for all x less than a, and returns a different value, say 10, for all x greater than or equal to a. f cannot have a derivative at a. If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h will be very steep, and as h tends to zero the slope tends to infinity. If h is positive, then a + h is on the high part of the step, so the secant line from a to a + h will have slope zero. Consequently the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.


The absolute value function is continuous, but fails to be differentiable at x = 0 since it has a sharp corner.However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function y = |x| is continuous at x = 0, but it is not differentiable there. If h is positive, then the slope of the secant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one. This can be seen graphically as a "kink" in the graph at x = 0. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance the function is not differentiable at x = 0.

Most functions which occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions which have a derivative at some point is a meager set in the space of all continuous functions.[citation needed] Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

The derivative as a function
Let f be a function that has a derivative at every point a in the domain of f. Because every point a has a derivative, there is a function which sends the point a to the derivative of f at a. This function is written f'(x) and is called the derivative function or the derivative of f. The derivative of f collects all the derivatives of f at all the points in the domain of f.

Sometimes f has a derivative at most, but not all, points of its domain. The function whose value at a equals f'(a) whenever f'(a) is defined and is undefined elsewhere is also called the derivative of f. It is still a function, but its domain is strictly smaller than the domain of f.

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions which have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D, then D(f) is the function f′(x). Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f′(a).

For comparison, consider the doubling function f(x) =2x; f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:

The operator D, however, is not defined on individual numbers. It is only defined on functions:

Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the squaring function,

D outputs the doubling function,

which we named f(x). This output function can then be evaluated to get f(1) = 2, f(2) = 4, and so on.

Higher derivatives
Let f be a differentiable function, and let f'(x) be its derivative. The derivative of f'(x) (if it has one) is written f''(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is written f'''(x) and is called the third derivative of f. These repeated derivatives are called higher-order derivatives.

A function f need not have a derivative, for example, if it is not continuous. Similarly, even if f does have a derivative, it may not have a second derivative. For example, let

.
An elementary calculation shows that f is a differentiable function whose derivative is

.
f'(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have k derivatives for any non-negative integer k but no (k + 1)-order derivative. A function that has k successive derivatives is called k times differentiable. If in addition the kth derivative is continuous, then the function is said to be of differentiability class Ck. (This is a stronger condition than having k derivatives. For an example, see differentiability class.) A function that has infinitely many derivatives is called infinitely differentiable or smooth.

On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.

The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example, if f is twice differentiable, then in the sense that If f is infinitely differentiable, then this is the beginning of the Taylor series for f.
Notations for differentiation
Main article: Notation for differentiation
Leibniz's notation
Main article: Leibniz's notation
The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when the equation y=f(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by Higher derivatives are expressed using the notation for the nth derivative of y=f(x) (with respect to x).

With Leibniz's notation, we can write the derivative of y at the point x=a in two different ways:

Leibniz's notation 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

Lagrange's notation
One of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark, so that the derivative of a function is denoted or simply . Similarly, the second and third derivatives are denoted and . Beyond this point, some authors use Roman numerals such as for the fourth derivative, whereas other authors place the number of derivatives in parentheses: in this case. The latter notation generalizes to yield the notation for the nth derivative of f.

Newton's notation
Main article: Newton's notation
Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a derivative. If y = f(t), then and denotes the first derivative of y with respect to t and denotes the second derivative. This notation is used almost exclusively for time derivatives, meaning that the independent variable of the function represents time. It is very common in physics and in mathematical disciplines connected with physics such as differential equations. While the notation becomes unmanageable for high-order derivatives, in practice only very few derivatives are needed.


Euler's notation
Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. The second derivative is denoted D2f, and the nth derivative is denoted Dnf.

If y=f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written Dxy or Dxf(x), although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

Euler's notation is useful for stating and solving linear differential equations.


Computing the derivative
The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. For some examples, see Derivative (examples). In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.


Rules for finding the derivative
Main article: Differentiation rules
In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. Some of the most basic rules are the following.

Constant rule: if f(x) is constant, then

Sum rule:
for all functions f and g and all real numbers a and b.
Product rule:
for all functions f and g.
Chain rule: If f(x) = h(g(x)), then
.
Derivatives of elementary functions
Main article: Table of derivatives
In addition, the derivatives of some common functions are useful to know.