 
Hello students, I am Gajendra Purohit and today I am here with a new topic
Partial Differentiation
and in this, we will study how to differentiate a function partially
Here, sometimes students have confusion that
how ordinary differentiation and partial differentiation are different from each other
So please have a look. I'll tell you the difference
'y' is a function of 'x' and, 'y' is a function of (x,t)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Let's say 'x' is variable and 't' is constant
 
 
 
Let's say here 'x' and 't' both are variable
 
 
 
 
 
 
If a function only depends on one variable then its derivative is known as as 'ordinary differentiation'
If a function depends on more than one variables then its derivative is known as as 'partial differentiation'
So students, if you are given a question to differentiate
then you have to find that which variable depends on whom
Here 'u' is depending on 'x', 'y' and 'z'
Therefore, its differentiation will be 'partial differentiation'
 
We will start with differentiating it with respect to 'z'
where 'x' and 'y' will be treated as constants
 
 
 
 
 
 
 
Now we will differentiate it with respect to 'y'
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now we will differentiate it with respect to 'x'
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
This is how we solve such type of questions
Now I'll take one or two more questions on this topic
Let's take two questions
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
'z' is depending on 'x' and 'y'
The variables in the denominator are always independent and the numerator depends on them.
Now let's differentiate it with respect to 'x' and, 'y' will remain constant
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The value of LHS is equal to the value of RHS
Hence proved.
This is how we prove this question
I'll take one last question and then this topic will come to an end.
 
 
 
We will start with differentiating it with respect to 'y'
and will take 'x' as a constant
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now we will differentiate it with respect to 'x'
 
 
 
 
 
 
 
 
 
 
 
 
 
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B.Sc and Engineering on topics like
Laplace and Fourier transformation, differential and integral calculus,
matrices, partial differential equation etc
 
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