Differentiation Formulas Explained with Rules, Tricks and Solved Examples

Differentiation in calculus refers to the process of finding the derivative of a function, which represents its rate of change concerning another variable. Together, differentiation and integration form the core concepts of calculus. Sir Isaac Newton laid the foundation for differential calculus, and its principles, including limits and derivatives, are widely applied across mathematics and science.

In mathematics, derivatives are essential for determining the maxima and minima of functions, finding the slope of curves, and identifying inflexion points. They are also used in practical applications such as calculating profit and loss in business through graphs, analysing temperature variations, and measuring speed or distance in units like miles per hour or kilometres per hour. The blog gives more information on the topic in detail:

What are Differentiation Formulas?

To understand differentiation formulas, we need to know differentiation and its derivates. Differentiation refers to the rate of change of one quantity concerning another. It represents the ratio of a small change in one variable to a small change in another dependent variable.

 If y = f(x) that is differentiable, then the differentiation is represented as f'(x) or dy/dx.

On the other hand, if a function y = f(x) is differentiable, its derivative is represented as f′(x) or dy/dx. The derivative of a real-valued function f(x), defined over an open interval I, is mathematically expressed as:

dy/dx = f′(x) = limΔx→0f(x+Δx)−f(x)Δx

The derivatives of elementary functions are commonly remembered as differentiation formulas.

Consider a function y = xn where n>0. Then,

f(x + Δx) = (x + Δx)n and f(x + Δx)-f(x) = (x + Δx)n - xn
f(x+δx)−f(x)/δx = limδx→0(x+δx)n −xn/ (x+δx)-x
=lim y→x yn−xn /y – x = nxn-1
where y = x + Δx and y → x as Δx → 0.

Differentiation Formula List

In differentiation all formulas, f′ and g′ represent the derivatives of functions f and g with respect to x. Both f and g are functions of x and are differentiated accordingly. The derivative of y with respect to x can also be expressed as dy/dx= Dxy

Basic Differentiation Formulas

• Power Rule: (d/dx) (xn ) = nxn-1
• Derivative of a Constant: (d/dx) (a) = 0
• Derivative of a Constant Multiplied by a Function f: (d/dx) (a. f) = af’
• Sum Rule: (d/dx) (f ± g) = f’ ± g’
• Product Rule: (d/dx) (fg)= fg’ + gf’ 
• Quotient Rule: d/dx (f/g) = gf’-fg’/g2

Trigonometric Differentiation Formulas

Trigonometry deals with the relationship between the angles and sides of triangles. It includes six primary ratios: sine, cosine, tangent, cotangent, secant, and cosecant. Basic trigonometric formulas are derived from these ratios. Below mentioned are the trigonometric differentiation formulas.

• d/dx* (sin x) = cos x
• d/dx* (cos x) = sin x
• d/dx* (tan x) = sec2 x
• d/dx* (cot x) = cosec2 x
• d/dx* (sec x) = sec x* tan x
• d/dx* (cosec x) = cosec x* cot x
• d/dx* (sinh x) = cosh x
• d/dx* (cosh x) = sinh x
• d/dx* (tanh x) = sech2 x
• d/dx* (coth x) = cosech2 x
• d/dx* (sech x) = sech x* tanh x
• d/dx* (cosech x) = cosech x* coth x

Logarithmic Differentiation Formulas

When a function is expressed as the product or quotient of multiple functions, such as y = f1 (x). y = f2 (x). f3 (x)….. g1 (x) g2 (x) …., logarithmic differentiation is used. The log differentiation formula involves taking the logarithm of the function and then differentiating it.

Similarly, if a function is in the form of one function raised to the power of another, like [f(x)]g(x) , we apply logarithmic differentiation by taking the natural logarithm (base e) before differentiating.

For example, if y = xx , then log y = x log x
1/y. dy/dx = log x + 1
dy/dx = y. (logx + 1)
= xx (logx + 1)

Way Forward

The concept of differentiation allows us to find out the rate of change of a function for a variable. It plays an important role in various mathematical and scientific applications. Differentiation formulas, including basic trigonometric and logarithmic differentiation rules, provide systematic methods for efficiently computing derivatives. This makes differentiation a basic instrument used in advanced studies and practice.

FAQs

Q1: What is differentiation?

A1: Differentiation is a technique used to determine how a function f(x) changes with respect to its input x. This rate of change is referred to as the derivative of f concerning x.

Q2: What is d/dx?

A2: d/dx is considered the derivative operator.

Q3: What are the basic rules of differentiation?

A3: The basic rules of differentiation are the power rule, sum rule, product rule, quotient rule, and chain rule, given by:

• Power Rule = d/dx(xn) = nxn-1
• Sum Rule = f'(x)=u'(x)±v'(x)
• Product Rule = f′(x) = u′(x) × v(x) + u(x) × v′(x)
• Quotient Rule = f’(x) = u’(x) . v(x) – u(x) . v’(x)/ (v(x))2
• Chain Rule = dy/dx = (dy/du) × (du/dx)