# 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.

## 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 𝛿y be an increment in y, corresponding to an increment 𝛿x in x. Then,

**y = f(x) and y + 𝛿y = f(x + 𝛿x) .**

**On subtraction, we get 𝛿y = f(x + 𝛿x) – 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.

- If f(x) = cos (x), then f'(x) = -sin x
- If f(x) = sin (x), then f'(x) = cos x
- If f(x) = tan (x), then f'(x) = sec
^{2}x - If f(x) = cot(x), then f'(x) = -cosec
^{2}x - If f(x) = ln(x), then f'(x) = 1/x
- If f(x) = e
^{x}, then f'(x) = e^{x} - If f(x) = x
^{n}, where n is any fraction or integer, then f'(x) = nx^{n-1} - 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(3x ^{3} – 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.