Derivatives

Definition

In derivatives, a dependent variable varies with respect to an independent variable. Derivatives are defined as the rate of change of one variable with respect to another. Derivatives are used when there is a change in quantity. Mathematically, they refer to an instantaneous rate of change of a quantity.

Derivatives are used to study the rate of change of quantities. For example, in physics, derivatives are used to study the rate of change of displacement, known as velocity, and rate of change of velocity, known as acceleration. Derivatives of displacement and velocity are denoted by:

Velocity = dx/dt, where x is the displacement, and t is the time.
Acceleration = dv/dt, where v is the velocity, and t is the time.

Derivatives in calculus

Derivatives are widely used in calculus, particularly differentiation. Differentiation is a measure of the rate of a variable y with respect to x. It is represented by f and denoted as:
f = dy/dx

For example, if x changes, then y will also change. This changes the entire function and is represented in the ratio form of dy/dx.
The inverse of differentiation is known as anti-differentiation.

Steps to find derivatives

1.) Let a slight change, say h, occur in the function x, so that the function becomes f (x + h).
2.) The change in function is– f (x + h) – f (x).
3.) The rate of function from x to (x + h) is denoted by:
    

Formulas of derivatives

• d/dx (k) = 0, where k is any constant
• d/dx(x) = 1
• d/dx(xn) = nxn-1
• d/dx (kx) = k, where k is any constant
• d/dx (√x) = 1/2√x
• d/dx (1/x) = -1/x2
• d/dx (log x) = 1/x, x > 0
• d/dx (ex) = ex
• d/ex (ax) = ax log a

Trigonometric derivatives

• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec2 x
• d/dx (cosec x) = -cosec x cot x
• d/dx (sec x) = sec x tan x
• d/dx (cot x) = -cosec2 x

Types of derivatives

There are two types of derivatives –

1. First-order derivative – The first order derivative tells us whether the function is increasing or decreasing. It is represented as an instantaneous rate of change of any quantity. The slope of the tangent line can predict the first-order derivative.
2. Second-order derivative – They are used to find out the shape of the graph of any given function. The graphs can be found out from their concavity, i.e. whether a graph is concave up or concave down.

Example 1: Find the derivatives of i) x6 ii) x4 + x3 – 5 iii) 1/x6
Solution:
1.) We know, the property of the derivative is d/dx (xn) = n xn-1
     Therefore, d/dx (x6) = 6 x5
2.) We know, the properties of the derivative are d/dx (xn) = n xn-1 and d/dx (k) = 0, where k is any constant. Here, the              constant is 5 since it does not have any coefficient in x.
     Therefore, d/dx x4 + x3 – 5 = (4x3 + 3x2 – 0
3.) We know, the property of the derivative is d/dx (xn) = n xn-1
     Therefore, f (x) = 1/x6 = d/dx x-6 = -6 x-7