Differentiation: definition and basic derivative rules

Differentiation

The process of differentiation informs the user of how much one quantity changes as a result of changes in another quantity. There are countless examples of derivatives.

For example, speed is the derivative of distance with respect to time. 

Differentiation is frequently used in applied sciences to model relationships between observed variables such as employment and inflation (Keynes).

A derivative can be thought of as the best possible approximation of an equation at a specified point. 

Note that the derivative will frequently change as the specified point changes – the differential will always remain the same. Differentiation works by approximating the slope of the tangent at each point of a function – the derivative is obtained by evaluating the differential at a specified point.

Differentiation is the process by which the derivative of a function is found. The opposite can be thought of as integration or anti-differentiation – the fundamental theory of calculus claims that these two terms describe identical processes. Anti-differentiation and differentiation are the 2 main operations in the field of calculus.

See also: Differentiation Formulas

What is Differentiation in Maths

In Mathematics, Within the field of calculus, a derivative identifies the rate at which output changes with respect to changes in the input.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy / dx

Derivative of  A Function

Let y = f(x) be a given continuous function. Then, the value of y depends upon the value of x and it changes with a change in the value of x. We use the word increment to denote a small change, i.e., an increase or decrease in the value of x and y.

Let ?y be an increment in y, corresponding to an increment ?x in x. Then,

y = f(x)  and y + ?y = f(x + ?x) .

On subtraction, we get ?y = f(x + ?x) – f(x).

The above limit, if it exists finitely is called the derivative or differential coefficient of y = f(x) with respect to x. It is denoted by

Differentiation Formulas

The list of most important Differentiation formulas are given below. Here, we assume f(x) is a function and f'(x) is the derivative of the function.

1. If f(x) = cos (x), then f'(x) = -sin x
2. If f(x) = sin (x), then f'(x) = cos x
3. If f(x) = tan (x), then f'(x) = sec ²x
4. If f(x) = cot(x), then f'(x) = -cosec ²x
5. If f(x) = ln(x), then f'(x) = 1/x
6. If f(x) = e ^x, then f'(x) = e ^x
7. If f(x) = x ⁿ, where n is any fraction or integer, then f'(x) = nx ^n-1
8. If f(x) = k, where k is a constant, then f'(x) = 0

Solved Examples of Differentiation

Q.1: Differentiate f(x) = 6x ⁴-7x+5 with respect to x.
Solution: Given: f(x) = 6x ⁴-7x+4

On differentiating both the sides w.r.t x, we get;

f'(x) = (4)(6)x ² – 9

f'(x) = 24 x ² – 9

This is the answer.

Q.2: Differentiate y = x(3x³ – 4)

Solution: Given, y = x(3x ³ – 4)

y = 3x ⁴ – 4x

On differentiating both the sides we get,

dy/dx = 9x ³ – 9

This is the answer.