First Partial Derivative

A partial derivative is a derivative that is used in Differential Geometry and vector calculus. In context with mathematics, sometimes the function depends on more than two variables.

Since the function depends on where the variable in derivatives converts it into a partial derivative, the partial derivative is a very important topic in vector calculus. Suppose a function is  f(x, y); this suggests that this function is dependent on both the variables y and x

But, where y and x are interrelated or dependent on each other. The derivative of f is known as the partial derivative, where f is differentiated concerning x and y remains constant, and again f is differentiated concerning y and x remains constant. This article will discuss first partial derivative examples, the interpretations of first-order partial derivatives and tables and contours to estimate partial derivatives.

What are Partial Derivatives?

Suppose a function f(x, y) is dependent on two variables, y, and x, where y and x are independent and independent of each other. Then it is said that the function f is

a partial function that depends on y and x. When f is differentiated between x and y, it stays constant, and again f is differentiated between y and x.

First Partial Derivative 

If the mathematical function U= f(x, y) and  f, or the  partial derivatives of f concerning x is denoted as ∂f/∂x and can be described as:

∂f/∂x = limδx→0 f(x + δx,y) – f(x,y)/δx

The solution for the partial derivative of f concerning Y then it can be denoted as ∂f/∂y and described as:

∂f/∂y = limδy→0 f(x,y + δy) – f(x,y)/δy

The equations, as mentioned earlier, are also known as the first  Partial derivative examples of f or partial derivatives of y x. When we calculate the first partial derivative examples of the mathematical function f concerning x, we consider y as a constant. The same goes for when we consider a mathematical function f concerning y and x remains constant.

There are also other types of derivatives,  namely:

•  Double partial derivative 

•  Mixed partial derivative

•  Higher-order partial derivative 

First Partial Derivative Examples

In the context of mathematics, there is a matrix known as the Jacobian matrix. The Jacobian Matrix can be defined as the matrix that comprises the first-order derivative of the first-order partial derivative of the vector functions.  The pseudo-inverse matrix of Jacobian can be utilised to solve inverse kinematics problems in robotics. 

Let us take another  look at it with the help of an example mentioned below:

f(a, b) = a3 + 4a + 5b

First, we will take b as a constant and determine ∂f/∂a concerning a.  ∂f/∂a also means the derivative or partial derivative of the function f(a, b).

∂f/∂a = b3 + 4 

The procedure of solving a partial derivative 

• ∂f/∂a (partial derivative) of b3a = b3a/a = b3

• ∂f/∂a (partial derivative) of 4a = 4a/a = 4

• ∂f/∂a (partial derivative) of 5b(constant term) = 0

• So therefore, ∂f/∂a = b3 + 4

Now we will take an as a constant and determine the value of b. ∂f/∂b intends the derivative of the partial derivative of the function f(a, b) concerning b.

∂f/∂b = 2b25 + 5

f(a, b) = b3 + 4a + 5

The procedure of solving a partial derivative 

• ∂f/∂b (partial derivative) of b3a = 2b3a

• ∂f/∂b (partial derivative) of 4a = 0 

• ∂f/∂a (partial derivative) of 5b = 5b/b = 5

• So therefore, ∂f/∂b = 2b2a + 5

Let us take another example:

Explicit function partial derivative:

b = 3a2 – 5c2 + 2a2c – 4ac2 – 9

To find ∂b/∂a.

The procedure of solving a partial derivative:

• ∂b/∂a (partial derivative) of 3a2 = 2(3a2–1) = 6a

• ∂b/∂a (partial derivative) of 5c2 = 0

• ∂b/∂a (partial derivative) of 2a2c = 2(2a2–1c) = 4ac

• ∂b/∂a (partial derivative) of 4ac2 = 4c2

• ∂b/∂a (partial derivative) of 9(constant term) = 0

• So therefore, ∂b/∂a = 6a+ 4ac – 4c2

Now to determine ∂b/∂c:

• ∂b/∂c (partial derivative) of 3a2 = 0

• ∂b/∂c (partial derivative) of 5c2 = 10 c

• ∂b/∂c (partial derivative) of 2a2c = 2a2

• ∂b/∂c (partial derivative) of 4ac2 = 8ac

• ∂b/∂c (partial derivative) of 9(constant term) = 0

• So therefore, ∂b/∂c = –10c + 2a2 – 8ac

Interpretations of First-Order Partial Derivatives 

The interpretations of first-order partial derivatives is a very short section that acknowledges the two main interpretations of derivatives or partial derivatives of a function which is a single variable and holds back partial derivatives with a small modification.

Example:

Calculate if the mathematical function f(a, b) = a3/b3  is decreasing or increasing at (2,5), if a is allowed to vary, and b is constant.

 Solution:

If a is allowed to vary and b is a constant mathematical function, fa(a, b)  and its value determined at the point can be written as:

f(a, b) = a3/b3  = fa (2,5) = 4/125 > 0

So, by the above equation, the partial derivative concerning a is positive, and b is constant, so the function f increases at the point (2,5)  when we vary (a).

Conclusion

The partial derivative is a very important topic in vector calculus. A partial derivative is a derivative that is used in Differential Geometry and vector calculus. In context with mathematics, sometimes the function depends on more than two variables. Suppose a function f(x, y) is dependent on two variables, y, and x, where y and x are independent and independent of each other. Then it is said that the function f is a partial function that depends on y and x. When f is differentiated between x and y, it stays constant, and again f is differentiated between y and x.