Differentiation and Integration: Overview, Questions, Preparation

In Calculus, differentiation and integration are quite important concepts. Integration proves to be useful while calculating the addition of discrete data, which you will find hard to add one-by-one or by using a single value. On the other hand, differentiation proves to be useful while observing the changes in the value of a particular quantity with respect to the change in the value of another quantity. 

For example, you can evaluate how speed changes according to the value of time, and this is a simple application of differentiation. Integration can be used to determine the length of the cable required to connect two points that are miles apart.

Differentiation

When the value of a function changes continuously as one of its variables keeps changing, it is known as differentiation. Graphically, it can be shown as a tangent’s slope over a particular point of a function. 

Mathematically, differentiation can be expressed as per the below expression:

f’(a) = (f (a+h) - f(a))/h where limit h tends to be 0 

If f(x) = y, where y is a function of x, then its derivative is expressed as dy/dx. It is also called as y’s derivative as per x. 

Integration

A method that is used to find the values of indefinite and definite integrals is called integration. If F(x) is referred to as an integration of f(x), it is given by the below expression:

∫ f(x) dx = F(x) + C

f(x) is referred to as an integrand. 

dx is the agent that is used for integration. 

C is the constant, and x is the variable.

• Integration and differentiation are processes that are opposite or inverse to each other.
• To determine the values of integration and differentiation, we will have to use the limits within which the function is defined. 
• Any function’s derivative is always unique, whereas the integrals of a function may or may not be unique. 

Important formulas

• d/dx (e^x) = e^x
• ∫ e^x dx = e^x + C
• d/dx (x) = 1
• ∫ a dx = ax + C
• d/dx cosx = -sinx
• d/dx sinx = cosx
• ∫cosx dx = sinx + C
• ∫sinx dx = -cosx + C
• d/dx a = 0  if a is a constant 
• ∫1dx = x + C
• d/dx tanx = sec²x
• ∫sec²x = tanx + C
• d/dx xⁿ = x^n-1
• ∫a dx = ax + C  

Details about Differentiation and Integration taught in Class XII

The topic of differentiation and integration is taught in Class XII’s chapter of ‘Integrals’. This chapter has a weightage of 12 marks, which means that this particular topic carries a weightage from 4 to 8 marks in the exam. 

1. y = x³ Find dy/dx 

Solution:

We have the formula if y = xⁿ, then dy/dx = nx^n-1

Therefore dy/dx = 3x²

2. If y = tan²x, find dy/dx

Solution:

dy/dx = 2 tanx x d/dx (tanx) = 2tanx x sec²x

3. Calculate dy/dx if y = 1 - tan²x/sec²x

Solution:

Now we have tan²x = sec²x - 1

Therefore, dy/dx = 1 - sec²x + 1/sec²x = 2 - sec²x/sec²x

Q: Give some applications of integration.

Q: Give some applications of differentiation.

Q: What is differential calculus?

Q: Which are the main divisions of calculus?

Q: What is the other name for integration?