Calculus was found by Isaac Newton and Gottfried Wilhelm Leibnitz in the 17th century as a study of Continuous change. It is a part of Mathematics, which is a study of derivatives, integrals, limits, functions, and the Taylor series. Calculus Derivatives is the change in a function and this function is related to the relationship between two variables so it’s the ratio of the differentials.

Different types of Calculus derivatives

Calculus has two main concept explained below:

1)Differential calculus
2)Integral Calculus

  • Differential Calculus

Differential Calculus is the change in a variable. The derivative ∂y/ ∂x is another function of x which can be differentiated. The Derivative of ∂y/ ∂x is also called the second derivative of y and is denoted by ∂^² y/ ∂^² 2. Similarly, in general, the Nth derivative of y is denoted by ∂^ny/ ∂x_n.

Problem 1:

If y=e^{ax}\sin {bx} ,
prove that y_2-2ay_1+(a^2+b^2)y=0

Solution:

We have, y=e^{ax}\sin {bx} \\

y_1= e^{ax}(\cos {bx.b} )+\sin {bx}(e^{ax}.a) =be^{ax}\cos bx+at \\

y_1-ay=be^{ax}\cos bx\\

Differentiation on both side

y_2-ay_1=be^{ax}(-\sin bx.b)+b\cos bx(e^{ax}.a)=-b^2y+a(y_1-ay)

As a result,y_2-2ay_1+(a^2+b^2)y=0
  • Integral Calculus

In Mathematics, Integral is the area under the graph of a function for some definite interval and is used to find the volume, area, Displacement, and other combining infinitesimal data. It is one of the main operations of Calculus and inverse operation. It is also referred as Anti-derivative or indefinite integral.

F(x)= \int f(x) dx

For example,

\int\sin^n x {\text dx} =-\frac{sin^{n-1} x cos x}{n}+\frac{n-1}{n}\int\sin^{n-2} x {\text dx}

Quotient Rule  in Calculus derivatives

Here, we will explain to you about the calculus quotient rule with an example:

This rule is a special rule in calculus derivates which is defined as a formal method of dividing the function of one differential to another.

Hence, we have mentioned some techniques to master this rule in calculus.

Calculus Quotient rule

if y=\frac{u}{v}

then, \frac{\text{d}y}{\text{d}x}=\frac{v\frac{\text{d}u}{\text{d}x}+u\frac{\text{d}v}{\text{d}x}}{v^2}

 

Problem 2: 

Solve using the quotient rule y=\frac{3}{x+1}

Solution:

We want to differentiate y=\frac{3}{x+1}
 
As per the rule we can see u = 3 and v = x+1.
 
So, the derivative of these two functions would be:
 
 \frac{\text{d}u}{\text{d}x}=0  and  \frac{\text{d}v}{\text{d}x}=1
 
Therefore, when we put this in Calculus quotient rule
 
\frac{\text{d}y}{\text{d}x}=\frac{(v \frac{\text{d}u}{\text{d}x}-u \frac{\text{d}v}{\text{d}x})}{v^2}
 
=\frac{((x+1)(0)-(3)(1))}{(x+1)^2}
 
=\frac{\text{(x+1)}{(0)}-\text{(3)}{(1)}}{\text{(x+1)}^2}
 
=\frac{-3}{(x+1)^2}
 
As a result, \frac{-3}{(x+1)^2} is the differential.

In Conclusion, We have learned about the integrals and derivatives Calculus with example that helped us to understand the changes between the values which are related by a function. Calculus Derivatives mainly focuses on two concepts of differential calculus which helps to find the rate of change of a quantity, whereas integral calculus helps to find the quantity when the rate of change is known. Therefore, this article will give you the basic knowledge of calculus derivates.

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