Differentiation: Definition
Differentiation is the process of determining a function’s derivative, or rate of change. In contrast to the abstract character of the theory that underpins it, differentiation can be accomplished using only algebraic manipulations, requiring only three basic derivatives, four rules of operation, and a working grasp of functions.
The three basic differentiation are:
(1) Algebraic functions –
| D(xn) = nxn − 1 |
where n is any real number
(2) Trigonometric functions –
| D(sin x) = cos x |
| D(cos x) = −sin x |
(3) Exponential functions –
| D(ex) = ex |
The theory offers the following basic rules for differentiating the sum, product, or quotient of the any two functions f(x) and g(x) whose derivatives are known (where a and b are constants) for functions built up of combinations of these types of functions:
Differentiation: Sum Rule
| D(af + bg) = aDf + bDg |
Differentiation: Product Rule
| D(fg) = fDg + gDf |
Differentiation: Quotient Rule
| D(f/g) = (gDf − fDg)/g2 |
Differentiation: Chain Rule
The chain rule is another basic rule that can be used to differentiate a composite function. If f(x) and g(x) are two functions, the composite function f(g(x)) is determined by first evaluating g(x) and then evaluating the function f at this g(x) value. The chain rule asserts that a product gives the derivative of a composite function:
| D(f(g(x))) = Df(g(x)) ∙ Dg(x) |
In other words, the first element on the right, Df(g(x)), indicates that the derivative of Df(x) is determined first as usual, and then x is replaced by the function g wherever it occurs (x).
Differentiation Formula of sec x, tan x, cot x, Log x
Some of the most important and basic formulae of differentiation are:
- If f(x) = xn, in which n is any fraction or integer, then
| f'(x) = nxn-1 |
- If f(x) = k, where k is a constant, then
| f'(x) = 0 |
- If f(x) = tan (x), then
| f'(x) = sec2x |
- If f(x) = cos (x), then
| f'(x) = -sin x |
- If f(x) = sin (x), then
| f'(x) = cos x |
- If f(x) = ln(x), then
| f'(x) = 1/x |
- If f(x) = ex, then
| f'(x) = ex |
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Differentiation: FAQs
What is differentiation used for?
In arithmetic, differentiation is used to calculate rates of change. The velocity, for example, is the rate of change of displacement (with respect to time) in mechanics. The acceleration is the rate of change of velocity (with respect to time).
How do you differentiate?
There are a few simple criteria that can be utilised to quickly distinguish between various functions. The derivative of y (with respect to x) is written dy/dx, pronounced “dee y by dee x” if y = some function of x (in other words, if y is equivalent to an expression including integers and x’s).
What is differentiation in simple words?
Differentiation is the process of determining a function’s derivative, or rate of change. In contrast to the abstract character of the theory that underpins it, differentiation can be accomplished using only algebraic manipulations, requiring only three basic derivatives, four rules of operation, and a working grasp of functions.
