Errors using partial differentiation. Today I show how I’ll calculate errors using partial differentiation. Have a look!!

**Errors using partial differentiation**

Suppose I have a function in two variables, say, . Now the error will be

Also, is the error in and is the error in .

Similarly, for a function in three variables, say, , the error will be

Also, is the error in is the error in and is the error in .

Now I’ll give some examples of how to use this formula.

**Solved examples of how to calculate errors using partial differentiation**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013)*, “The deflection at the center of a rod is known to be given by , where is a constant. If increases by percent, by percent, and decreases by percent, find the percentage increase in .”

**Solution**

Now in this example, the deflection at the center of a rod is

(1)

with as a constant.

And I have to find out the percentage increase in when increases by percent, by percent, and decreases by percent.

So I’ll start with the partial differentiation of .

**Step 1**

First of all, I’ll differentiate in equation (1) partially with respect to .

Since is a function of three variables, and , I’ll use the same technique as the * first-order partial derivative of functions with three variables*.

So that means

Now I’ll differentiate equation (1) partially with respect to to get the value of as

Next, I’ll differentiate equation (1) partially with respect to to get the value of as

Now I’ll get the percentage increase in .

**Step 2**

As I can see, the variable increases by percent. So that means

Similarly, the variable increases by percent. So that means

But the variable decreases by percent. And that gives

Therefore, according to the formula mentioned above, the percentage increase in is

Now I’ll substitute all the values which I have got so far or are given in the example already, in this formula.

And that means

Next, I’ll simplify it. So that gives

Also, from equation (1), I already know that . Thus I can say that

Now I’ll simplify it to get

Therefore the increase in is percent. Hence I can conclude that this is the answer to the given example.

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2013)*, “The coefficient of rigidity of a wire of length and uniform diameter is given by , where is a constant. If errors percent and percent are possible in measuring and respectively, determine the maximum percentage error in the calculated value of .”

**Solution**

Now in this example, the coefficient of rigidity of a wire of length and uniform diameter is

(2)

with as a constant.

And the task is to get the maximum percentage error in the calculated value of . Also, the possible errors are percent in measuring and percent in measuring .

So, I’ll start with the partial differentiation of .

**Step 1**

First of all, I’ll differentiate in equation (2) partially with respect to .

Since is a function of two variables, and , I’ll use the same technique as the **First-order partial derivative of functions with two variables.**

So that means

Now I’ll differentiate equation (2) partially with respect to to get the value of as

Now I’ll get the maximum percentage error in the calculated value of .

**Step 2**

As I can see, the maximum possible error in measuring the variable is percent. So that means

Similarly, the maximum possible error in measuring the variable is percent. So that means

Therefore, according to the formula mentioned above, maximum possible error in measuring is

Now I’ll substitute all the values which I have got so far or are given in the example already, in this formula.

And that means

Now I’ll simplify it. So that gives

Also, from equation (2), I already know that . Thus I can say that

Now I’ll simplify it to get

Therefore the maximum percentage error in the calculated value of is percent. Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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