Differentiation formulas

 Introduction to Differentiation Formulas

The differentiation formulas are used in maths to find how fast something is changing. For example, if a car travels a distance s = t² ,we can use differentiation formulas to find out its speed at any time t. In simple words, these formulas help us know the rate of change of 1 quality with respect to another.

There are many important rules in differentiation formulas, such as the power rule, product rule, quotient rule, chain rule, & trigonometry formulas. These rules are used not only in maths but also in real-life situations. For example, we can find how quickly water fills a tank or how fast the temperature is rising or falling.

With the help of step-by-step explanations and solved examples, students can easily understand how to apply these formulas in different types of problems. With practice, you will not only do well in exams but also understand how maths works in real life, like speed, growth, or change over time.

 

Table of Contents

• Definition of Differentiation Formulas
• Differentiation formula list
• Power law
• Derivative of a constant
• Derivative of a constant multiplied by a function
• Sum Rule
• Product Rule
• Quotient Rule
• Differentiation Formulas for Trigonometric Functions
• Differentiation Formulas for Inverse Trigonometric Functions
• Application of differentiation formulas
• Solved Examples
• Practice Questions
• Conclusion
• FAQs on Differentiation Formulas

 

Definition of Differentiation Formulas

Differentiation formulas are a set of rules in mathematics that help us find the rate of change of one quantity to another. In simple words, they show how quickly a function is increasing or decreasing when its variable changes.

For example, if the side of a square is x, then its area is A = a². By using differentiation, we can find how quickly the area of the square increases when the side length increases.

Differentiation formula list

In all formulas below, f' and f' derivatives (rate of change) of the functions f and g for X. This means that we find out how the function changes when x changes occur. We can also write the derivative of y as dy/dx.

There are some common and useful differentiation formulas:

• Power law: (d/dx) (xⁿ) = nx^n-1
  Multiply by power and reduce power from 1

• Derivative of a constant: (d/dx) (a) = 0
  A number without x always has a derivative of 0
  

• Derivative of a constant multiplied by a function: (d/dx) (a . f) = af'
  Keep the constant outside and differentiate the function

• Sum Rule: (d/dx) (f ± g) = f' ± g'
  Differentiate each term separately

• Product Rule: (d/dx) (fg) = fg' + gf'
  First, keep the first function and differentiate the second, then keep the second and differentiate the first, and add them. 

• Quotient Rule: d/dx (f/g) = (gf' - fg')/g2
  Bottom times derivative of top minus top times derivative of bottom, divided by bottom squared

 

Differentiation Formulas for Trigonometric Functions

Trigonometry is a branch of mathematics that studies the relationship between the angles and sides of triangles. We have already learned about the six most important trigonometric ratios: sine (SIN), cosine (COS), tangent (TAN), cotangent (COT), secant (SEC), and cosecant (COSEC). These functions are widely used in mathematics, physics, and engineering.

In calculation, we can find out how these trigonometric functions change as the angle changes. This process is called 'differentiation,' and the results are called 'derivatives'. For example, when we differentiate sin 𝑥, we get cos 𝑥. These formulas are very useful for solving problems related to waves, speed, electricity, and many other areas.

Below is a list of differentiation formulas for trigonometric and hyperbolic functions:

• The derivative of sine is cosine: d/dx (sin x) = cos x

• The derivative of cosine is negative sine: d/dx (cos x) = – sin x

• The derivative of tangent is secant squared: d/dx (tan x) = sec² x

• The derivative of cotangent is negative cosecant squared: d/dx (cot x) =  –csc² x

• The derivative of secant is secant times tangent: d/dx (sec x) = sec x . tan x

• The derivative of cosecant is negative cosecant times cotangent: d/dx (csc x)= –cscx . cot x

• The derivative of hyperbolic sine is hyperbolic cosine: d/dx (sinh x) = cosh x

• The derivative of hyperbolic cosine is hyperbolic sine: d/dx (cosh x) = sinh x

• The derivative of hyperbolic tangent is hyperbolic secant squared: d/dx (tanh x) = sech² x

• The derivative of hyperbolic cotangent is negative hyperbolic cosecant squared: d/dx (coth x) = –cosech² x

• The derivative of hyperbolic secant is negative hyperbolic secant times hyperbolic tangent: d/dx (sech x) = –sech x . tanh x

 

Differentiation Formulas for Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse of the usual trigonometric functions you have already studied. While general trigonometric functions (e.g., sin, cos, tan) take an angle and give you a ratio, inverse trigonometric functions take a ratio and give you an angle.

For example:

• If  , Then sin 0 = (1/2)  ⇒  0 = sin^-1 (1/2).

• Here  is the inverse of the sine function, which tells us if the angle is (1/2).

In calculus, we study how these inverse trigonometric functions change when the value of 𝑥 changes. This process is called differentiation, and it helps to solve problems in mathematics, physics, and engineering, especially in integration, coordinate geometry, and wave speed problems.

Below is a list of differentiation formulas for inverse trigonometric functions, as well as what they mean:

1. The derivative of inverse sine: d/dx (sin^-1x) = 1 / (√ 1- x²) 

2. The derivative of inverse cosine: the negative sign means the function decreases as 𝑥 increases: d/dx (cos^-1 x) =  – (1 / √ 1- x²)

3. The derivative of inverse tangent: d/dx (tan^-1x) = 1 / (√ 1 + x²) )

4. The derivative of inverse cotangent: d/dx (cot^-1x) = – (1 / 1+ x²)

5. The derivative of inverse secant, the absolute value |𝑥|, is used to ensure the denominator is always positive: d/dx (sec^-1x) = (1 / |x| √ x²– 1 )

6. The derivative of inverse cosecant, the negative sign again shows a decreasing function: d/dx (csc^-1x) = – (1 / |x| √ x²– 1 )

These formulas are important because they are used directly to solve many problems, especially in integration, differential equations, and geometry problems involving angles.

 

Application of differentiation formulas

Discrimination formulas are not only used in mathematics problems, but also in many real-life situations. They help us know how quickly something is increasing or decreasing.

1. Speed and Distance: We can find how fast an object is moving forward if we know the distance it covers in time.

2. Growth and Change: They help us know how fast things grow or shrink, such as the area of a shape when the side increases, or how the height of a plant changes over time.

3. Physics and Nature: Differentiation is used to find out how fast the water fills a tank, how quickly the temperature changes, or how quickly an object falls to the ground.

4. Economics and Business: It can be used to check how the profits or costs change when the number of items produced or sold changes.

 

Solved Examples

By solving Differentiation Questions, you can easily build the conceptual fluency required for exams and competitive tests.

1. Power Rule: d/dx (x³)

Ans:

• d/dx (xⁿ) = nx^n-1
• d/dx  (x³) = 3x²

2. Constant Rule: d/dx (7) 

Ans: The derivative of any constant is 0.

3. Constant × Function: d/dx (5x⁴)

Ans: 

• Keep the constant; differentiate the function.
• 5 . (4x³) = 20 x³

4. Sum/Difference + Power Rule: d/dx( 3x² – 4x + 1 )

Ans:

• Derivative of 3x^2– 3.2x^2-1 = 6x
• Derivative of  – 4x → – 4.1x^1-1 = – 4
• Derivative of constant 1 → 0
• So, d/dx( 3x² – 4x + 1 ) = 6x – 4.

5. Power Rule with Fraction: d/dx (√ x)

Ans:

• √ x = x^(1/2)
• 1/2 x^-(1/2) 
• 1 / 2√ x

 

Practice Questions

1. Differentiate: f(x) =  x⁵ + 2x³ – 7x + 4 

2. Find d/dx of sin(x) . ln(x)

3. Differentiate: [ x (2 + 1) / (x + 2) ]

4. Use chain: f(x) = the whole root of 3x² + 2x 

5. Apply product rule: f(x) = x² . e^x

 

Conclusion

Differentiation helps us to find out how quickly things change. And learning power rules, product rules, quotient rules, chain rules, and trigonometric rules makes it easier to solve problems. With practice, you can quickly find the slopes and rate of change and solve real-life problems in speed, growth, heat, and more. Understand the rules, solve step by step, and practise regularly, then differentiation will look simple and useful.

 

FAQs on Differentiation Formulas

1. What are the basic formulas of differentiation?

Ans: 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) = nx^n-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)

• Chain Rule = dy/dx = (dy/du) * (du/dx)

2. What is the differentiation of 25?

Ans: Since 25 is constant with respect to x , the derivative of 25 with respect to x is 0.

3. What is a derivative formula?

Ans: A derivative helps us to know the changing relationship between two variables. Mathematically, the derivative formula is helpful to find the slope of a line, to find the slope of a curve, and to find the change in one measurement with respect to another measurement. The derivative formula is (d/dx) (xⁿ) = n . x^n-1.

4. What are the 7 differentiation rules?

Ans: Types of Rules of Differentiation

• The Constant Rule.

• The Constant Multiplier Rule.

• The Sum and Difference Rule.

• The Power Rule.

• The Exponential Rule.

• The Product Rule.

• The Quotient Rule.

• The Chain Rule.

5. What are the 5 basic integration formulas?

Ans: The list of basic integral formulas is given below:

• ∫ 1 dx = x + C.

• ∫ a dx = ax + C.

• ∫ xⁿ dx = ((xⁿ⁺¹) / (n+1)) +C ; n≠1.

• ∫ sin x dx = – cos x + C.

• ∫ cos x dx = sin x + C.

• ∫ sec²x dx = tan x + C.

• ∫ csc²x dx = -cot x + C.

• ∫ sec x (tan x) dx = sec x + C.